For an integer n∈Zn∈Zn inZn \in \mathbb{Z}n∈Z, let [n][n][n][n][n] denote the multiplication-by- nnnnn endomorphism of AAAAA. Then LLL\mathscr{L}L is called even or symmetric if there is an isomorphism [−1]∗L≅L[−1]∗L≅L[-1]^(**)L~=L[-1]^{*} \mathscr{L} \cong \mathscr{L}[−1]∗L≅L. It is called odd or antisymmetric if [−1]∗L≅L⊗(−1)[−1]∗L≅L⊗(−1)[-1]^(**)L~=Lox(-1)[-1]^{*} \mathscr{L} \cong \mathscr{L} \otimes(-1)[−1]∗L≅L⊗(−1). If LLL\mathscr{L}L is any ample invertible sheaf on AAAAA, then L⊗[−1]∗LL⊗[−1]∗LLox[-1]^(**)L\mathscr{L} \otimes[-1]^{*} \mathscr{L}L⊗[−1]∗L is ample and even. So any abelian variety admits an even, ample invertible sheaf.
Suppose that LLL\mathscr{L}L is even. Then [2]* L≅L⊗4L≅L⊗4L~=L^(ox4)\mathscr{L} \cong \mathscr{L}^{\otimes 4}L≅L⊗4 is a consequence of the Theorem of the Cube. So Theorem 2.4 implies hA,L∘[2]=4hA,LhA,L∘[2]=4hA,Lh_(A,L)@[2]=4h_(A,L)h_{A, \mathscr{L}} \circ[2]=4 h_{A, \mathscr{L}}hA,L∘[2]=4hA,L as classes and by iteration hA,L∘[2k]=hA,L∘2k=h_(A,L)@[2^(k)]=h_{A, \mathscr{L}} \circ\left[2^{k}\right]=hA,L∘[2k]=4khA,L4khA,L4^(k)h_(A,L)4^{k} h_{A, \mathscr{L}}4khA,L for all k≥1k≥1k >= 1k \geq 1k≥1. We fix a representative hA,L′hA,L′h_(A,L)^(')h_{A, \mathscr{L}}^{\prime}hA,L′ of hA,LhA,Lh_(A,L)h_{A, \mathscr{L}}hA,L and find hA,L′∘[2k]=hA,L′∘2k=h_(A,L)^(')@[2^(k)]=h_{A, \mathscr{L}}^{\prime} \circ\left[2^{k}\right]=hA,L′∘[2k]=4khA,L′+Ok(1)4khA,L′+Ok(1)4^(k)h_(A,L)^(')+O_(k)(1)4^{k} h_{A, \mathscr{L}}^{\prime}+O_{k}(1)4khA,L′+Ok(1) on A(Q¯)A(Q¯)A( bar(Q))A(\overline{\mathbb{Q}})A(Q¯). Tate's Limit Argument is used to show convergence in the following definition.
Definition 2.6. Let LLL\mathscr{L}L be an even invertible sheaf on AAAAA and let P∈A(Q¯)P∈A(Q¯)P in A( bar(Q))P \in A(\overline{\mathbb{Q}})P∈A(Q¯). Then the limit
If LLL\mathscr{L}L is even, then (2.3) immediately implies h^A,L([2](P))=4h^A,L(P)h^A,L([2](P))=4h^A,L(P)hat(h)_(A,L)([2](P))=4 hat(h)_(A,L)(P)\hat{h}_{A, \mathscr{L}}([2](P))=4 \hat{h}_{A, \mathscr{L}}(P)h^A,L([2](P))=4h^A,L(P) for all P∈A(Q¯)P∈A(Q¯)P in A( bar(Q))P \in A(\overline{\mathbb{Q}})P∈A(Q¯). If PPPPP has finite order, then [2m](P)=[2n](P)2m(P)=2n(P)[2^(m)](P)=[2^(n)](P)\left[2^{m}\right](P)=\left[2^{n}\right](P)[2m](P)=[2n](P) for distinct integers 0≤m<n0≤m<n0 <= m < n0 \leq m<n0≤m<n by the Pigeonhole Principle. Thus h^A,L(P)=0h^A,L(P)=0hat(h)_(A,L)(P)=0\hat{h}_{A, \mathscr{L}}(P)=0h^A,L(P)=0.
There is nothing special about [2]. Indeed, one can replace [2] by [m[m[m[\mathrm{m}[m ] in (2.3) for any integer m≥2m≥2m >= 2m \geq 2m≥2; one then needs to replace 4k4k4^(k)4^{k}4k in the denominator by m2km2km^(2k)m^{2 k}m2k.
What happens if LLL\mathscr{L}L is an odd invertible sheaf? In this case, [2] ∗L≅L⊗2∗L≅L⊗2^(**)L~=L^(ox2){ }^{*} \mathscr{L} \cong \mathscr{L}^{\otimes 2}∗L≅L⊗2. Then a similar limit (2.3) exists, but now we need to divide by 2k2k2^(k)2^{k}2k.
The set of odd invertible sheaves is a divisible subgroup of Pic(A)Picâ¡(A)Pic(A)\operatorname{Pic}(A)Picâ¡(A). From this, one can show that, after possibly extending the base field FFFFF, any invertible sheaf LLL\mathscr{L}L on AAAAA decom-
poses as L+⊗L−L+⊗L−L_(+)oxL_(-)\mathscr{L}_{+} \otimes \mathscr{L}_{-}L+⊗L−with L+L+L_(+)\mathscr{L}_{+}L+even and LLL\mathscr{L}L - odd. One then defines h^A,L=h^A,L++h^A,Lh^A,L=h^A,L++h^A,Lhat(h)_(A,L)= hat(h)_(A,L_(+))+ hat(h)_(A,L)\hat{h}_{A, \mathscr{L}}=\hat{h}_{A, \mathscr{L}_{+}}+\hat{h}_{A, \mathscr{L}}h^A,L=h^A,L++h^A,L; the decomposition of LLL\mathscr{L}L is not quite unique, but this ambiguity does not affect h^A,Lh^A,Lhat(h)_(A,L)\hat{h}_{A, \mathscr{L}}h^A,L.
For our purposes, we often restrict to even invertible sheaves.
Theorem 2.7. Let us keep the notation above. In particular, AAAAA is an abelian variety defined over a number field F⊆Q¯F⊆Q¯F sube bar(Q)F \subseteq \overline{\mathbb{Q}}F⊆Q¯.
(i) Then association L↦h^A,LL↦h^A,LL|-> hat(h)_(A,L)\mathscr{L} \mapsto \hat{h}_{A, \mathscr{L}}L↦h^A,L is a group homomorphism from Pic(V)Picâ¡(V)Pic(V)\operatorname{Pic}(V)Picâ¡(V) to the additive group of real-valued maps A(Q¯)→RA(Q¯)→RA( bar(Q))rarrRA(\overline{\mathbb{Q}}) \rightarrow \mathbb{R}A(Q¯)→R.
Suppose LLL\mathscr{L}L is an invertible sheaf on AAAAA.
holds for all P,Q∈A(Q¯)P,Q∈A(Q¯)P,Q in A( bar(Q))P, Q \in A(\overline{\mathbb{Q}})P,Q∈A(Q¯).
(iv) If LLL\mathscr{L}L is even and ample, then h^A,Lh^A,Lhat(h)_(A,L)\hat{h}_{A, \mathscr{L}}h^A,L takes nonnegative values and vanishes precisely on Ators Ators A_("tors ")A_{\text {tors }}Ators .
(v) If LLL\mathscr{L}L is even and ample, then h^A,Lh^A,Lhat(h)_(A,L)\hat{h}_{A, \mathscr{L}}h^A,L induces a well-defined map A(Q¯)⊗R→A(Q¯)⊗R→A( bar(Q))oxRrarrA(\overline{\mathbb{Q}}) \otimes \mathbb{R} \rightarrowA(Q¯)⊗R→[0,∞)[0,∞)[0,oo)[0, \infty)[0,∞). It is the square of a norm ∥⋅∥∥⋅∥||*||\|\cdot\|∥⋅∥ on the RRR\mathbb{R}R-vector space A(Q¯)⊗RA(Q¯)⊗RA( bar(Q))oxRA(\overline{\mathbb{Q}}) \otimes \mathbb{R}A(Q¯)⊗R and satisfies the parallelogram equality.
The norm ∥⋅∥∥⋅∥||*||\|\cdot\|∥⋅∥ allows us to do geometry in the RRR\mathbb{R}R-vector space A(Q¯)⊗RA(Q¯)⊗RA( bar(Q))oxRA(\overline{\mathbb{Q}}) \otimes \mathbb{R}A(Q¯)⊗R (which is infinite dimensional if dimA≥1)dimâ¡A≥1)dim A >= 1)\operatorname{dim} A \geq 1)dimâ¡A≥1). Indeed, for z,w∈A(Q¯)⊗Rz,w∈A(Q¯)⊗Rz,w in A( bar(Q))oxRz, w \in A(\overline{\mathbb{Q}}) \otimes \mathbb{R}z,w∈A(Q¯)⊗R, we define
The Mordell-Weil Theorem implies that A(F)⊗RA(F)⊗RA(F)oxRA(F) \otimes \mathbb{R}A(F)⊗R is finite dimensional. We will see that ∥⋅∥∥⋅∥||*||\|\cdot\|∥⋅∥ is a suitable norm to do Euclidean geometry in A(F)⊗RA(F)⊗RA(F)oxRA(F) \otimes \mathbb{R}A(F)⊗R.
3. VOJTA'S APPROACH TO THE MORDELL CONJECTURE
Recall that the Mordell Conjecture was proved first by Faltings. In this section we briefly describe Vojta's approach to the Mordell Conjecture [62]. At the core is the deep Vojta inequality which we state here for a curve in an abelian variety.
Let AAAAA be an abelian variety defined over a number field F⊆Q¯F⊆Q¯F sube bar(Q)F \subseteq \overline{\mathbb{Q}}F⊆Q¯. Let LLL\mathscr{L}L be an ample and even invertible sheaf on AAAAA. We write ∥⋅∥=h^A,L1/2∥⋅∥=h^A,L1/2||*||= hat(h)_(A,L)^(1//2)\|\cdot\|=\hat{h}_{A, \mathscr{L}}^{1 / 2}∥⋅∥=h^A,L1/2 for the norm on A(Q¯)⊗RA(Q¯)⊗RA( bar(Q))oxRA(\overline{\mathbb{Q}}) \otimes \mathbb{R}A(Q¯)⊗R defined in Theorem 2.7.
Theorem 3.1 (Vojta's inequality). Let C⊆AC⊆AC sube AC \subseteq AC⊆A be a curve that is defined over FFFFF and that is not a translate of an algebraic subgroup of A. There are c1>1,c2>1c1>1,c2>1c_(1) > 1,c_(2) > 1c_{1}>1, c_{2}>1c1>1,c2>1, and c3>0c3>0c_(3) > 0c_{3}>0c3>0 with the following property. If P,Q∈C(Q¯)P,Q∈C(Q¯)P,Q in C( bar(Q))P, Q \in C(\overline{\mathbb{Q}})P,Q∈C(Q¯) satisfy
The values c1,c2,c3c1,c2,c3c_(1),c_(2),c_(3)c_{1}, c_{2}, c_{3}c1,c2,c3 depend on the curve CCCCC. One remarkable aspect is that Vojta's inequality is a statement about pairs of Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯-points of the curve CCCCC. So all three values c1,c2,c3c1,c2,c3c_(1),c_(2),c_(3)c_{1}, c_{2}, c_{3}c1,c2,c3 are "absolute," i.e., we can take them as independent of the base field FFFFF of AAAAA and CCCCC. Both c1c1c_(1)c_{1}c1 and c2c2c_(2)c_{2}c2 are of "geometric nature." They depend only on the degree of CCCCC with respect to LLL\mathscr{L}L and other discrete data attached to AAAAA and CCCCC. In contrast, c3c3c_(3)c_{3}c3 is of "arithmetic nature." Roughly speaking, it depends on suitable heights of coefficients that define the curve CCCCC in some projective embedding.
Let us now sketch a proof of Mordell's Conjecture using the Vojta inequality and the classical Mordell-Weil Theorem.
Suppose CCCCC has genus g≥2g≥2g >= 2g \geq 2g≥2. Without loss of generality, C(F)≠∅C(F)≠∅C(F)!=O/C(F) \neq \emptysetC(F)≠∅. So we fix a base point P0∈C(F)P0∈C(F)P_(0)in C(F)P_{0} \in C(F)P0∈C(F), then P↦P−P0P↦P−P0P|->P-P_(0)P \mapsto P-P_{0}P↦P−P0 induces an immersion C→Jac(C)C→Jacâ¡(C)C rarr Jac(C)C \rightarrow \operatorname{Jac}(C)C→Jacâ¡(C). So we may assume that CCCCC is a curve inside the ggggg-dimensional A=Jac(C)A=Jacâ¡(C)A=Jac(C)A=\operatorname{Jac}(C)A=Jacâ¡(C). Note that CCCCC is not a translate of an algebraic subgroup of its Jacobian since g≥2g≥2g >= 2g \geq 2g≥2.
By the finiteness statement around (3.1), it suffices to show that there are at most finitely many large points.
For any z∈Jac(C)(F)⊗Rz∈Jacâ¡(C)(F)⊗Rz in Jac(C)(F)oxRz \in \operatorname{Jac}(C)(F) \otimes \mathbb{R}z∈Jacâ¡(C)(F)⊗R, we define the truncated cone
Any large point in C(F)C(F)C(F)C(F)C(F) has image in some T(zj)TzjT(z_(j))T\left(z_{j}\right)T(zj) from above. After possibly adjusting NNNNN, one can arrange that each zjzjz_(j)z_{j}zj is the image of a point Pj∈C(F)Pj∈C(F)P_(j)in C(F)P_{j} \in C(F)Pj∈C(F) with ∥Pj∥2>c3Pj2>c3||P_(j)||^(2) > c_(3)\left\|P_{j}\right\|^{2}>c_{3}∥Pj∥2>c3 for all j∈{1,…,N}j∈{1,…,N}j in{1,dots,N}j \in\{1, \ldots, N\}j∈{1,…,N}. If Q∈C(F)Q∈C(F)Q in C(F)Q \in C(F)Q∈C(F) has image in T(zj)TzjT(z_(j))T\left(z_{j}\right)T(zj), then Vojta's inequality implies h^L(Q)1/2=∥Q∥≤c2∥Pj∥h^L(Q)1/2=∥Q∥≤c2Pjhat(h)_(L)(Q)^(1//2)=||Q|| <= c_(2)||P_(j)||\hat{h}_{\mathscr{L}}(Q)^{1 / 2}=\|Q\| \leq c_{2}\left\|P_{j}\right\|h^L(Q)1/2=∥Q∥≤c2∥Pj∥. But then Q∈C(F)Q∈C(F)Q in C(F)Q \in C(F)Q∈C(F) lies in a finite ball as in (3.1). So the number of possible QQQQQ that come to lie in a single T(zj)TzjT(z_(j))T\left(z_{j}\right)T(zj) is finite. Thus C(F)C(F)C(F)C(F)C(F) is finite.
The constants c1,c2c1,c2c_(1),c_(2)c_{1}, c_{2}c1,c2, and c3c3c_(3)c_{3}c3 in Vojta's inequality can be made effective in terms of AAAAA and CCCCC. Yet, the proof as a whole is ineffective. Indeed, the height bound for QQQQQ depends on the hypothetical point PjPjP_(j)P_{j}Pj. However, there is no guarantee that PjPjP_(j)P_{j}Pj exists and if it does not, there is no known way to know for sure.
Using Mumford's Gap Principle, one can show that the number of large points C(F)C(F)C(F)C(F)C(F) that come to lie in a single T(zj)TzjT(z_(j))T\left(z_{j}\right)T(zj) is bounded from above by c′⋅crkJac(C)(F)c′⋅crkJac(C)(F)c^(')*c^(rkJac(C)(F))c^{\prime} \cdot c^{\mathrm{rkJac}(C)(F)}c′⋅crkJac(C)(F), after possibly increasing the constants. Now we need to introduce dependency on c2c2c_(2)c_{2}c2. But the base ccccc will remain geometric in nature, it depends on the genus of ggggg. But it does not depend on c3c3c_(3)c_{3}c3 or other arithmetic properties of CCCCC that encode the heights of coefficients defining the said curve. Finally, as observed by Bombieri, 7 is admissible for ccccc for any genus. Indeed, he showed that 4 is admissible for c1c1c_(1)c_{1}c1.
Recall that Vojta's inequality with the same values of c1,c2,c3c1,c2,c3c_(1),c_(2),c_(3)c_{1}, c_{2}, c_{3}c1,c2,c3 applies to points in C(F′)CF′C(F^('))C\left(F^{\prime}\right)C(F′) for all finite extensions F′/FF′/FF^(')//FF^{\prime} / FF′/F. The upshot is that the number of large points of C(F′)CF′C(F^('))C\left(F^{\prime}\right)C(F′) is bounded by
where c,c′c,c′c,c^(')c, c^{\prime}c,c′ depend on CCCCC, but not on F′F′F^(')F^{\prime}F′.
The dichotomy between large and moderate points was already visible in Vojta's work. But its origin is older and already appears in modified form in work of Thue, Siegel, Mahler, and Roth on diophantine approximation.
With our eyes set on Mazur's question, we aim to obtain good bounds for the number of moderate points. In the coming two sections we explain our general approach to the proof of Theorem 1.12.
Having worked with a fixed abelian variety in Sections 2.2 and 3, we now shift gears and work in a family of abelian varieties.
Example 4.1. Let Y(2)=P1∖{0,1,∞}Y(2)=P1∖{0,1,∞}Y(2)=P^(1)\\{0,1,oo}Y(2)=\mathbb{P}^{1} \backslash\{0,1, \infty\}Y(2)=P1∖{0,1,∞}. For λ∈Y(2)(C)λ∈Y(2)(C)lambda in Y(2)(C)\lambda \in Y(2)(\mathbb{C})λ∈Y(2)(C), we have an elliptic curve Eλ⊆P2Eλ⊆P2E_(lambda)subeP^(2)\mathcal{E}_{\lambda} \subseteq \mathbb{P}^{2}Eλ⊆P2 determined by
where the origin is [0:1:0][0:1:0][0:1:0][0: 1: 0][0:1:0]. The total space EEE\mathcal{E}E is a surface presented with a closed immersion E↪P2×Y(2)E↪P2×Y(2)E↪P^(2)xx Y(2)\mathcal{E} \hookrightarrow \mathbb{P}^{2} \times Y(2)E↪P2×Y(2). It is called the Legendre family of elliptic curves and is an abelian scheme over Y(2)Y(2)Y(2)Y(2)Y(2). So we can add two complex points of EEE\mathscr{E}E if they are in the same fiber above Y(2)Y(2)Y(2)Y(2)Y(2). More precisely, there is an addition morphism E×SE→EE×SE→EExx_(S)ErarrE\mathcal{E} \times_{S} \mathcal{E} \rightarrow \mathcal{E}E×SE→E over SSSSS, as well as an inversion morphism E→EE→EErarrE\mathcal{E} \rightarrow \mathcal{E}E→E over SSSSS. Finally, the zero section of EEE\mathcal{E}E is given by λ↦([0:1:0],λ)λ↦([0:1:0],λ)lambda|->([0:1:0],lambda)\lambda \mapsto([0: 1: 0], \lambda)λ↦([0:1:0],λ).
Consider a geometrically irreducible smooth quasiprojective variety SSSSS defined over a number field F⊆Q¯F⊆Q¯F sube bar(Q)F \subseteq \overline{\mathbb{Q}}F⊆Q¯. Let π:A→SÏ€:A→Spi:Ararr S\pi: \mathcal{A} \rightarrow SÏ€:A→S be an abelian scheme over SSSSS. So each fiber As=π−1(s)As=π−1(s)A_(s)=pi^(-1)(s)\mathcal{A}_{s}=\pi^{-1}(s)As=π−1(s), where s∈S(Q¯)s∈S(Q¯)s in S( bar(Q))s \in S(\overline{\mathbb{Q}})s∈S(Q¯), is an abelian variety. We have an addition morphism on the fibered square A××SA→AA××SA→AAxxxx_(S)ArarrA\mathcal{A} \times \times_{S} \mathcal{A} \rightarrow \mathcal{A}A××SA→A and an inversion morphism A→AA→AArarrA\mathscr{A} \rightarrow \mathcal{A}A→A; both are relative over SSSSS. Addition induces a multiplication-by- nnnnn morphism [n]:A→A[n]:A→A[n]:ArarrA[n]: \mathcal{A} \rightarrow \mathcal{A}[n]:A→A over SSSSS for all n∈Zn∈Zn inZn \in \mathbb{Z}n∈Z.
Let s∈S(Q¯)s∈S(Q¯)s in S( bar(Q))s \in S(\overline{\mathbb{Q}})s∈S(Q¯). Then AsAsA_(s)\mathscr{A}_{s}As is an abelian variety in PnPnP^(n)\mathbb{P}^{n}Pn. We have two functions, h^A|As(Q¯)h^AAs(Q¯)hat(h)_(A)|_(A_(s)( bar(Q)))\left.\hat{h}_{\mathcal{A}}\right|_{\mathcal{A}_{s}(\overline{\mathbb{Q}})}h^A|As(Q¯) and h|As(Q¯)hAs(Q¯)h|_(A_(s)( bar(Q)))\left.h\right|_{\mathcal{A}_{s}(\overline{\mathbb{Q}})}h|As(Q¯); the latter is the restriction of the Weil height on PnPnP^(n)\mathbb{P}^{n}Pn. By Theorem 2.7(ii), their difference is bounded in absolute value in function of sssss.
exists. Suppose, in addition, that the geometric generic fiber of A→SA→SArarr S\mathcal{A} \rightarrow SA→S has trivial trace over Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯. Then the limit vanishes if and only if VVVVV is an irreducible component of ker[N]kerâ¡[N]ker[N]\operatorname{ker}[N]kerâ¡[N] for some N≥1N≥1N >= 1N \geq 1N≥1. Otherwise the limit is positive.
Motivated by Theorem 4.2, the author showed the next theorem. It may serve as a higher-dimensional substitute for Silverman's Theorem 4.2. For an irreducible subvariety of VVVVV of AAA\mathcal{A}A that dominates SSSSS, we write Vη¯Vη¯V_( bar(eta))V_{\bar{\eta}}Vη¯ for the geometric generic fiber of π|V:V→SÏ€V:V→Spi|_(V):V rarr S\left.\pi\right|_{V}: V \rightarrow SÏ€|V:V→S. This is a possibly reducible subvariety of the geometric generic fiber Aη¯Aη¯A_( bar(eta))\mathcal{A}_{\bar{\eta}}Aη¯ of A→SA→SArarr S\mathcal{A} \rightarrow SA→S.
Theorem 4.3 ([35]). Suppose S=Y(2)S=Y(2)S=Y(2)S=Y(2)S=Y(2) and let A=E[g]A=E[g]A=E[g]\mathcal{A}=\mathcal{E}[g]A=E[g] be the ggggg-fold fibered power of the Legendre family of elliptic curves. Suppose V⊆E[g]V⊆E[g]V subeE^([g])V \subseteq \mathcal{E}^{[g]}V⊆E[g] dominates Y(2)Y(2)Y(2)Y(2)Y(2) and
Vη¯Vη¯V_( bar(eta))V_{\bar{\eta}}Vη¯ is not a finite union of irreducible components of algebraic subgroups of Aη¯Aη¯A_( bar(eta))\mathcal{A}_{\bar{\eta}}Aη¯.
Then there exist c(V)>0c(V)>0c(V) > 0c(V)>0c(V)>0 and a Zariski open and dense subset U⊆VU⊆VU sube VU \subseteq VU⊆V with
(4.4)hY(2)(π(P))≤c(V)max{1,h^A(P)} for all P∈U(Q¯)(4.4)hY(2)(Ï€(P))≤c(V)max1,h^A(P) for all P∈U(Q¯){:(4.4)h_(Y(2))(pi(P)) <= c(V)max{1, hat(h)_(A)(P)}quad" for all "P in U( bar(Q)):}\begin{equation*}
h_{Y(2)}(\pi(P)) \leq c(V) \max \left\{1, \hat{h}_{\mathcal{A}}(P)\right\} \quad \text { for all } P \in U(\overline{\mathbb{Q}}) \tag{4.4}
\end{equation*}(4.4)hY(2)(π(P))≤c(V)max{1,h^A(P)} for all P∈U(Q¯)
The Zariski open UUUUU cannot in general be taken to equal VVVVV. But there is a natural description of this set in geometric terms through unlikely intersections.
The hypothesis (4.3) is necessary and essentially rules out that VVVVV itself is a family of abelian subvarieties.
Gao and the author [33] then generalized Theorem 4.3 to an abelian scheme when the base is again a smooth curve SSSSS defined over Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯. Here more care is needed in connection
with the hypothesis (4.3). Indeed, if A=A×SA=A×SA=A xx S\mathcal{A}=A \times SA=A×S is a constant abelian scheme, where AAAAA is an abelian variety, then (4.4) cannot hold generically for V=Y×SV=Y×SV=Y xx SV=Y \times SV=Y×S. Roughly speaking, the condition in [33] that replaces (4.3) also needs to take into account a possible constant part of Aη¯Aη¯A_( bar(eta))\mathscr{A}_{\bar{\eta}}Aη¯. If Aη¯Aη¯A_( bar(eta))\mathscr{A}_{\bar{\eta}}Aη¯ has no constant part, i.e., if its Q¯(η¯)/Q¯Q¯(η¯)/Q¯bar(Q)( bar(eta))// bar(Q)\overline{\mathbb{Q}}(\bar{\eta}) / \overline{\mathbb{Q}}Q¯(η¯)/Q¯-trace is 0 , then (4.3) suffices for SSSSS a curve. The case of a higher-dimensional base requires even more care, as we will see.
There were two applications of the height bound in [33].
First, and in the same paper, we proved new cases of the geometric Bogomolov Conjecture for an abelian variety defined over the function field of the curve SSSSS. This approach relied on Silverman's Theorem 4.2. It was used earlier in [35] to give a new proof of the Geometric Bogomolov Conjecture in a power of an elliptic curve. The number field case of the Bogomolov Conjecture was proved by Ullmo [61] and Zhang [71] in the 1990s. Progress in the function field case was later made by Cinkir, Faber, Moriwaki, Gubler, and Yamaki. For the state of the Geometric Bogomolov Conjecture as of 2017, we refer to a survey of Yamaki [64]. Gubler's strategy works in arbitrary characteristic and was expanded on by Yamaki. In joint work [10] with Cantat, Gao, and Xie, the author later established the Geometric Bogomolov Conjecture in characteristic 0 by bypassing the height inequality (4.4). Very recently, Xie and Yuan [63] announced a proof of the Geometric Bogomolov Conjecture in arbitrary characteristic. Their approach builds on the work of Gubler and Yamaki.
Second, and in later joint work with Dimitrov and Gao [23], we established uniformity for the number of rational points in the spirit of Mazur's question for curves parametrized by the 1-dimensional base SSSSS.
As we shall see, the proof of Theorem 1.12 requires a height comparison result like (4.4) for abelian schemes over a base SSSSS of any dimension. But now the correct condition to impose on VVVVV is more sophisticated and cannot be easily read off of the geometric generic fiber as in (4.3). The condition relies on the Betti map, which we introduce in the next section.
4.1. Degenerate subvarieties and the Betti map
In this section, SSSSS is a smooth irreducible quasiprojective variety over CCC\mathbb{C}C. Let π:A→SÏ€:A→Spi:Ararr S\pi: \mathcal{A} \rightarrow SÏ€:A→S again be an abelian scheme over SSSSS of relative dimension g≥1g≥1g >= 1g \geq 1g≥1.
For each s∈S(C)s∈S(C)s in S(C)s \in S(\mathbb{C})s∈S(C), the fiber As(C)As(C)A_(s)(C)\mathcal{A}_{s}(\mathbb{C})As(C) is a complex torus of dimension ggggg. Forgetting the complex structure, each ggggg-dimensional complex torus is diffeomorphic to (R/Z)2g(R/Z)2g(R//Z)^(2g)(\mathbb{R} / \mathbb{Z})^{2 g}(R/Z)2g as a real Lie group. By Ehresmann's Theorem, this diffeomorphism extends locally in the analytic topology on the base. That is, there is a contractible open neighborhood UUUUU of sssss in S(C)S(C)S(C)S(\mathbb{C})S(C) and a diffeomorphism AU=π−1(U)→(R/Z)2g×UAU=π−1(U)→(R/Z)2g×UA_(U)=pi^(-1)(U)rarr(R//Z)^(2g)xx U\mathscr{A}_{U}=\pi^{-1}(U) \rightarrow(\mathbb{R} / \mathbb{Z})^{2 g} \times UAU=π−1(U)→(R/Z)2g×U over UUUUU. Fiberwise this diffeomorphism can be arranged to be a group isomorphism above each point of UUUUU. Thus we can locally trivialize the abelian scheme at the cost of sacrificing the complex-analytic structure.
The trivialization is not entirely unique as we can let a matrix in GL2g(Z)GL2g(Z)GL_(2g)(Z)\mathrm{GL}_{2 g}(\mathbb{Z})GL2g(Z) act in the natural way on the real torus (R/Z)2g(R/Z)2g(R//Z)^(2g)(\mathbb{R} / \mathbb{Z})^{2 g}(R/Z)2g. But since UUUUU is connected, this is the only ambiguity. It is harmless for what follows.
The Betti map βUβUbeta_(U)\beta_{U}βU attached to UUUUU is the composition of the trivialization followed by the projection
(i) For all s∈Us∈Us in Us \in Us∈U, the restriction βU|As(C):As(C)→(R/Z)2gβUAs(C):As(C)→(R/Z)2gbeta_(U)|_(A_(s)(C)):A_(s)(C)rarr(R//Z)^(2g)\left.\beta_{U}\right|_{\mathcal{A}_{s}(\mathbb{C})}: \mathscr{A}_{s}(\mathbb{C}) \rightarrow(\mathbb{R} / \mathbb{Z})^{2 g}βU|As(C):As(C)→(R/Z)2g is a diffeomorphism of real Lie groups. In particular, P∈AUP∈AUP inA_(U)P \in \mathcal{A}_{U}P∈AU has finite order in its respective fiber if and only if βU(P)∈(Q/Z)2gβU(P)∈(Q/Z)2gbeta_(U)(P)in(Q//Z)^(2g)\beta_{U}(P) \in(\mathbb{Q} / \mathbb{Z})^{2 g}βU(P)∈(Q/Z)2g.
(ii) For all P∈UP∈UP in UP \in UP∈U the fiber βU−1(βU(P))βU−1βU(P)beta_(U)^(-1)(beta_(U)(P))\beta_{U}^{-1}\left(\beta_{U}(P)\right)βU−1(βU(P)) is a complex-analytic subset of AUAUA_(U)\mathcal{A}_{U}AU.
Definition 4.4. An irreducible closed subvariety V⊆AV⊆AV subeAV \subseteq \mathcal{A}V⊆A that dominates SSSSS is called degenerate if for all UUUUU and βUβUbeta_(U)\beta_{U}βU as above and all smooth points PPPPP of VU=π|V−1(U)VU=Ï€V−1(U)V_(U)= pi|_(V)^(-1)(U)V_{U}=\left.\pi\right|_{V} ^{-1}(U)VU=Ï€|V−1(U) the differential of dP(βU|VU)dPβUVUd_(P)(beta_(U)|_(V_(U)))\mathrm{d}_{P}\left(\left.\beta_{U}\right|_{V_{U}}\right)dP(βU|VU) satisfies
(4.5)rkdP(βU|VU)<2dimV(4.5)rkdPâ¡Î²UVU<2dimâ¡V{:(4.5)rkd_(P)(beta_(U)|_(V_(U))) < 2dim V:}\begin{equation*}
\operatorname{rkd}_{P}\left(\left.\beta_{U}\right|_{V_{U}}\right)<2 \operatorname{dim} V \tag{4.5}
\end{equation*}(4.5)rkdPâ¡(βU|VU)<2dimâ¡V
It has become customary to call VVVVV degenerate if it is not nondegenerate.
For all smooth points PPPPP of VUVUV_(U)V_{U}VU, the left-hand side of (4.5) is at most the right-hand side, which equals the real dimension of VUVUV_(U)V_{U}VU. It is also at most 2g2g2g2 g2g, the real dimension of a fiber of A→SA→SArarr S\mathcal{A} \rightarrow SA→S. Moreover, if the maximal rank of dβUdβUdbeta_(U)\mathrm{d} \beta_{U}dβU on VUVUV_(U)V_{U}VU is attained at PPPPP then the maximal rank is attained also in a neighborhood of PPPPP in VUVUV_(U)V_{U}VU. Being nondegenerate is a local property.
Let us consider some examples.
Example 4.5. (i) If SSSSS is a point, then AAA\mathscr{A}A is an abelian variety and an arbitrary subvariety V⊆AV⊆AV subeAV \subseteq \mathcal{A}V⊆A is nondegenerate because βSβSbeta_(S)\beta_{S}βS is a diffeomorphism.
(ii) Suppose dimV>gdimâ¡V>gdim V > g\operatorname{dim} V>gdimâ¡V>g. Then rkdP(βU∣VU)≤2g<2dimVrkdPâ¡Î²U∣VU≤2g<2dimâ¡Vrkd_(P)(beta_(U)∣V_(U)) <= 2g < 2dim V\operatorname{rkd}_{P}\left(\beta_{U} \mid V_{U}\right) \leq 2 g<2 \operatorname{dim} VrkdPâ¡(βU∣VU)≤2g<2dimâ¡V for all smooth PPPPP and so VVVVV is degenerate. In particular, AAA\mathscr{A}A is a degenerate subvariety of AAA\mathscr{A}A if dimS≥1dimâ¡S≥1dim S >= 1\operatorname{dim} S \geq 1dimâ¡S≥1.
(iii) Suppose A=A×SA=A×SA=A xx S\mathscr{A}=A \times SA=A×S is a constant abelian scheme with AAAAA an abelian variety. If Y⊆AY⊆AY sube AY \subseteq AY⊆A is a closed irreducible subvariety and if dimS≥1dimâ¡S≥1dim S >= 1\operatorname{dim} S \geq 1dimâ¡S≥1, then Y×SY×SY xx SY \times SY×S is degenerate. Indeed, the rank is at most 2dimY<2dimY×S2dimâ¡Y<2dimâ¡Y×S2dim Y < 2dim Y xx S2 \operatorname{dim} Y<2 \operatorname{dim} Y \times S2dimâ¡Y<2dimâ¡Y×S.
(iv) Suppose VVVVV is an irreducible component of ker[N]kerâ¡[N]ker[N]\operatorname{ker}[N]kerâ¡[N] for some integer N≥1N≥1N >= 1N \geq 1N≥1. Any point in V(C)V(C)V(C)V(\mathbb{C})V(C) has order dividing NNNNN (and, in fact, equal to NNNNN ). So the image of βU|VUβUVUbeta_(U)|_(V_(U))\left.\beta_{U}\right|_{V_{U}}βU|VU is finite and hence VVVVV is degenerate if dimS≥1dimâ¡S≥1dim S >= 1\operatorname{dim} S \geq 1dimâ¡S≥1.
(v) Suppose VVVVV is the image of a section S→AS→AS rarrAS \rightarrow \mathcal{A}S→A. If the geometric generic fiber of A→SA→SArarr S\mathcal{A} \rightarrow SA→S has trivial trace, then (βU)|VUβUVU(beta_(U))|_(V_(U))\left.\left(\beta_{U}\right)\right|_{V_{U}}(βU)|VU is constant if and only if VVVVV is an irreducible component of ker[N]kerâ¡[N]ker[N]\operatorname{ker}[N]kerâ¡[N] for some N≥1N≥1N >= 1N \geq 1N≥1. This is Manin's Theorem of the Kernel, we refer to Bertrand's article [7] for the history of this theorem.
(vi) Suppose A=E[g]A=E[g]A=E^([g])\mathcal{A}=\mathcal{E}^{[g]}A=E[g] and VVVVV are as in Theorem 4.3. One step in the proof of this theorem consisted in verifying that VVVVV, subject to hypothesis (4.3), is nondegenerate. Crucial input came from the monodromy action of the fundamental group of the base Y(2)=P1∖{0,1,∞}Y(2)=P1∖{0,1,∞}Y(2)=P^(1)\\{0,1,oo}Y(2)=\mathbb{P}^{1} \backslash\{0,1, \infty\}Y(2)=P1∖{0,1,∞} on the first homology of a fiber AsAsA_(s)\mathcal{A}_{s}As with sssss in general position. In this case the monodromy action is unipotent at the cusps 0 and 1 of YYYYY (2). This enabled the author to use a result of Kronecker from diophantine approximation. Already Masser and Zannier [44] used the monodromy action in their earlier work for VVVVV a curve.
(vii) If SSSSS is a curve, then the monodromy action of the fundamental group of S(C)S(C)S(C)S(\mathbb{C})S(C) on the homology of fibers of A→SA→SArarr S\mathscr{A} \rightarrow SA→S is locally quasiunipotent. But if SSSSS is projective, then there are no cusps. So exploiting monodromy in this setting required a different approach. In [33] Gao and the author used o-minimal geometry and the Pila-Wilkie Counting Theorem [51]. A related case was solved by Cantat, Gao, and Xie in collaboration with the author [10]; we used dynamical methods.
We assume that AAA\mathscr{A}A carries symplectic level- ℓâ„“â„“\ellâ„“ structure for some fixed ℓ≥3ℓ≥3â„“ >= 3\ell \geq 3ℓ≥3 and that LLL\mathscr{L}L induces a principal polarization. For the proof of Theorem 1.12, it suffices to have the following height bound under these conditions. We also refer to [24, THEOREM B.1] for a version that relaxes some of the conditions.
Theorem 4.6 ([24, THEOREM 1.6]). Let VVVVV be a nondegenerate irreducible subvariety of AAA\mathcal{A}A that dominates SSSSS. There exist c(V)>0,c′(V)≥0c(V)>0,c′(V)≥0c(V) > 0,c^(')(V) >= 0c(V)>0, c^{\prime}(V) \geq 0c(V)>0,c′(V)≥0, and a Zariski open and dense subset U⊆VU⊆VU sube VU \subseteq VU⊆V with
hS(π(P))≤c(V)h^A(P)+c′(V) for all P∈U(Q¯)hS(Ï€(P))≤c(V)h^A(P)+c′(V) for all P∈U(Q¯)h_(S)(pi(P)) <= c(V) hat(h)_(A)(P)+c^(')(V)quad" for all "P in U( bar(Q))h_{S}(\pi(P)) \leq c(V) \hat{h}_{\mathcal{A}}(P)+c^{\prime}(V) \quad \text { for all } P \in U(\overline{\mathbb{Q}})hS(Ï€(P))≤c(V)h^A(P)+c′(V) for all P∈U(Q¯)
We refer to Yuan and Zhang's Theorem 6.2.2 [67] for a height inequality in the dynamical setting.
The positive constant c(V)c(V)c(V)c(V)c(V) in Theorem 4.6 ultimately comes from the application of Siu's Criterion. As such it can expressed in geometric terms.
5. APPLICATION TO MODERATE POINTS ON CURVES
In this section we sketch the main lines of the proof of Theorem 1.12. It will be enough to bound the number of moderate points, see Section 3.
5.1. The Faltings-Zhang morphism
Smooth curves of genus g≥2g≥2g >= 2g \geq 2g≥2 defined over Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯ are classified by the Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯-points of a quasiprojective variety, the coarse moduli space. For us it is convenient to work with symplectic level- ℓâ„“â„“\ellâ„“ structure on the Jacobian for some fixed integer ℓ≥3ℓ≥3â„“ >= 3\ell \geq 3ℓ≥3. With this extra data, we obtain a fine moduli space MgMgM_(g)\mathbb{M}_{g}Mg, together with a universal family Cg→MgCg→MgC_(g)rarrM_(g)\mathfrak{C}_{g} \rightarrow \mathbb{M}_{g}Cg→Mg. Fibers of this family are smooth curves of genus ggggg with the said level structure on the Jacobian. Then MgMgM_(g)\mathbb{M}_{g}Mg carries the structure of a smooth quasiprojective variety of dimension 3g−33g−33g-33 g-33g−3 defined over a cyclotomic field. For convenience, we replace MgMgM_(g)\mathbb{M}_{g}Mg by an irreducible component by choosing a complex root of unity of order ℓâ„“â„“\ellâ„“ and consider it as defined over Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯.
The Torelli morphism τ:Mg→AgÏ„:Mg→Agtau:M_(g)rarrA_(g)\tau: \mathbb{M}_{g} \rightarrow \mathbb{A}_{g}Ï„:Mg→Ag takes a smooth curve to its Jacobian with the level structure; here AgAgA_(g)\mathbb{A}_{g}Ag denotes the fine moduli space of ggggg-dimensional abelian varieties with a principal polarization and symplectic level −ℓ−ℓ-â„“-\ell−ℓ structure.
Let M≥0M≥0M >= 0M \geq 0M≥0 be an integer and consider M+1M+1M+1M+1M+1 points P0,…,PM∈Cg(C)P0,…,PM∈Cg(C)P_(0),dots,P_(M)inC_(g)(C)P_{0}, \ldots, P_{M} \in \mathfrak{C}_{g}(\mathbb{C})P0,…,PM∈Cg(C) in the same fiber CCCCC of Cg→MgCg→MgC_(g)rarrM_(g)\mathfrak{C}_{g} \rightarrow \mathbb{M}_{g}Cg→Mg. The differences [P1]−[P0],…,[PM]−[P0]P1−P0,…,PM−P0[P_(1)]-[P_(0)],dots,[P_(M)]-[P_(0)]\left[P_{1}\right]-\left[P_{0}\right], \ldots,\left[P_{M}\right]-\left[P_{0}\right][P1]−[P0],…,[PM]−[P0] are divisors of degree 0 on CCCCC. We obtain MMMMM complex points in the Jacobian of CCCCC and so MMMMM complex points of VgVgV_(g)\mathfrak{V}_{g}Vg. We obtain a commutative diagram
of morphisms of schemes; here the exponent [M][M][M][M][M] denotes taking the MMMMM th fibered power over the base. The morphism DDD\mathscr{D}D is called the Faltings-Zhang morphism; see [26, LEMMA 4.1] and [71, LEMMA 3.1] for important applications to diophantine geometry of variants of this morphism. The morphism DDD\mathscr{D}D is proper.
A modified version of this construction is also useful. Say S→MgS→MgS rarrM_(g)S \rightarrow \mathbb{M}_{g}S→Mg is a quasifinite morphism with SSSSS an irreducible quasiprojective variety defined over Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯. We obtain a proper morphism D:Cg[M+1]×MgS→Vg[M]×AgSD:Cg[M+1]×MgS→Vg[M]×AgSD:C_(g)^([M+1])xx_(M_(g))S rarrV_(g)^([M])xx_(A_(g))S\mathscr{D}: \mathfrak{C}_{g}^{[M+1]} \times_{\mathbb{M}_{g}} S \rightarrow \mathscr{V}_{g}^{[M]} \times_{\mathbb{A}_{g}} SD:Cg[M+1]×MgS→Vg[M]×AgS, again called Faltings-Zhang morphism.
Gao, using his Ax-Schanuel Theorem for the universal family NgNgN_(g)\mathfrak{N}_{g}Ng [30] and a characterization [28] of bialgebraic subvarieties of ℜgℜgℜ_(g)\mathfrak{\Re}_{g}ℜg, obtained
Theorem 5.1 (Gao [29]). Let S→MgS→MgS rarrM_(g)S \rightarrow \mathbb{M}_{g}S→Mg be as above, i.e., a quasifinite morphism from an irreducible quasiprojective variety SSSSS defined over Q¯Q¯bar(Q)\overline{\mathbb{Q}}Q¯ and g≥2g≥2g >= 2g \geq 2g≥2. If M≥dimMg+1=M≥dimâ¡Mg+1=M >= dim M_(g)+1=M \geq \operatorname{dim} \mathbb{M}_{g}+1=M≥dimâ¡Mg+1=3g−23g−23g-23 g-23g−2, then D(Eg[M+1]×MgS)DEg[M+1]×MgSD(E_(g)^([M+1])xx_(M_(g))S)\mathscr{D}\left(\mathfrak{E}_{g}^{[M+1]} \times_{\mathbb{M}_{g}} S\right)D(Eg[M+1]×MgS) is a nondegenerate subvariety of Ig[M]×AgSIg[M]×AgSI_(g)^([M])xx_(A_(g))S\mathfrak{I}_{g}^{[M]} \times_{\mathbb{A}_{g}} SIg[M]×AgS.
Mok, Pila, and Tsimerman [47] earlier proved an Ax-Schanuel Theorem for Shimura varieties. Gao's result [30] is a "mixed" version in the abelian setting. We refer to the survey [4] on recent developments in functional transcendence.
The hypothesis g≥2g≥2g >= 2g \geq 2g≥2 is crucial. The definition of the Faltings-Zhang morphism makes sense for g=1g=1g=1g=1g=1. But it will be surjective and the image is degenerate expect in the (for our purposes uninteresting) case dimS=0dimâ¡S=0dim S=0\operatorname{dim} S=0dimâ¡S=0.
We consider here for simplicity only the case S=MgS=MgS=M_(g)S=\mathbb{M}_{g}S=Mg.
Using basic dimension theory, we see dimD(Sg[M+1])≤M+1+dimMgdimâ¡DSg[M+1]≤M+1+dimâ¡Mgdim D(S_(g)^([M+1])) <= M+1+dim M_(g)\operatorname{dim} \mathscr{D}\left(\mathfrak{S}_{g}^{[M+1]}\right) \leq M+1+\operatorname{dim} \mathbb{M}_{g}dimâ¡D(Sg[M+1])≤M+1+dimâ¡Mg. The image lies in the fibered power Vg[M]Vg[M]V_(g)^([M])\mathfrak{V}_{g}^{[M]}Vg[M] where the relative dimension is MgMgMgM gMg. A necessary condition for D(Cg[M+1])DCg[M+1]D(C_(g)^([M+1]))\mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right)D(Cg[M+1]) to be nondegenerate is dimD(Cg[M+1])≤Mgdimâ¡DCg[M+1]≤Mgdim D(C_(g)^([M+1])) <= Mg\operatorname{dim} \mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right) \leq M gdimâ¡D(Cg[M+1])≤Mg, see Example 4.5(ii). This inequality follows from
(5.1)M+3g−2=M+1+dimMg≤Mg(5.1)M+3g−2=M+1+dimâ¡Mg≤Mg{:(5.1)M+3g-2=M+1+dim M_(g) <= Mg:}\begin{equation*}
M+3 g-2=M+1+\operatorname{dim} \mathbb{M}_{g} \leq M g \tag{5.1}
\end{equation*}(5.1)M+3g−2=M+1+dimâ¡Mg≤Mg
If M≤3M≤3M <= 3M \leq 3M≤3, the numerical condition (5.1) is not satisfied for any g≥2g≥2g >= 2g \geq 2g≥2. For this reason, we cannot hope to work with the image of Cg×MgCgCg×MgCgC_(g)xx_(M_(g))C_(g)\mathfrak{C}_{g} \times_{\mathbb{M}_{g}} \mathfrak{C}_{g}Cg×MgCg in VgVgVg\mathfrak{V} gVg by taking differences. Moreover, there seems to be no reasonable way to work with a single copy of CgCgC_(g)\mathfrak{C}_{g}Cg, where the relations between dimensions would be even worse. The numerical condition (5.1) is satisfied for all M≥4M≥4M >= 4M \geq 4M≥4 and all g≥2g≥2g >= 2g \geq 2g≥2. Gao's Theorem implies that M≥3g−2M≥3g−2M >= 3g-2M \geq 3 g-2M≥3g−2 is sufficient to guarantee nondegeneracy.
We can thus apply Theorem 4.6 to the image D(Cg[M+1])DCg[M+1]D(C_(g)^([M+1]))\mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right)D(Cg[M+1]) of the Faltings-Zhang morphism in Vg[M]×AgMgVg[M]×AgMgV_(g)^([M])xx_(A_(g))M_(g)\mathscr{V}_{g}^{[M]} \times_{\mathbb{A}_{g}} \mathbb{M}_{g}Vg[M]×AgMg. Let M=3g−2M=3g−2M=3g-2M=3 g-2M=3g−2, then
for all (P0,…,PM)∈U(Q¯)P0,…,PM∈U(Q¯)(P_(0),dots,P_(M))in U( bar(Q))\left(P_{0}, \ldots, P_{M}\right) \in U(\overline{\mathbb{Q}})(P0,…,PM)∈U(Q¯) above s∈Mg(Q¯)s∈Mg(Q¯)s inM_(g)( bar(Q))s \in \mathbb{M}_{g}(\overline{\mathbb{Q}})s∈Mg(Q¯) where UUUUU is a Zariski open and dense subset of D(Cg[M+1])DCg[M+1]D(C_(g)^([M+1]))\mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right)D(Cg[M+1]). The constants c(g)>0c(g)>0c(g) > 0c(g)>0c(g)>0 and c′(g)≥0c′(g)≥0c^(')(g) >= 0c^{\prime}(g) \geq 0c′(g)≥0 depend on the various choices made regarding projective immersions of MgMgM_(g)\mathbb{M}_{g}Mg and NlgNlgNl_(g)\mathfrak{N l}_{g}Nlg. Ultimately, they depend only on ggggg once these choices have been made.
The Zariski open UUUUU cannot be replaced by D(Cg[M+1])DCg[M+1]D(C_(g)^([M+1]))\mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right)D(Cg[M+1]). Indeed, the right-hand side of (5.2) vanishes on the diagonal P0=P1=⋯=PMP0=P1=⋯=PMP_(0)=P_(1)=cdots=P_(M)P_{0}=P_{1}=\cdots=P_{M}P0=P1=⋯=PM whereas the left-hand side is unbounded as sssss varies.
As 2M=6g−42M=6g−42M=6g-42 M=6 g-42M=6g−4 we find
(5.4)hMg(s)≤c(g)(6g−4)max1≤j≤M∥Pj−P0∥2 for all (P0,…,PM)∈U(Q¯)(5.4)hMg(s)≤c(g)(6g−4)max1≤j≤M Pj−P02 for all P0,…,PM∈U(Q¯){:(5.4)h_(M_(g))(s) <= c(g)(6g-4)max_(1 <= j <= M)||P_(j)-P_(0)||^(2)quad" for all "(P_(0),dots,P_(M))in U( bar(Q)):}\begin{equation*}
h_{\mathbb{M}_{g}}(s) \leq c(g)(6 g-4) \max _{1 \leq j \leq M}\left\|P_{j}-P_{0}\right\|^{2} \quad \text { for all }\left(P_{0}, \ldots, P_{M}\right) \in U(\overline{\mathbb{Q}}) \tag{5.4}
\end{equation*}(5.4)hMg(s)≤c(g)(6g−4)max1≤j≤M∥Pj−P0∥2 for all (P0,…,PM)∈U(Q¯)
Morally, (5.4) states that among a ( 3g−1)3g−1)3g-1)3 g-1)3g−1)-tuple of points on a curve of genus ggggg in general position, there must be a pair that repels one another with respect to the norm ∥⋅∥∥⋅∥||*||\|\cdot\|∥⋅∥. The squared distance of such a pair is larger than a positive multiple, depending only on ggggg, of the modular height hMg(s)hMg(s)h_(M_(g))(s)h_{\mathbb{M}_{g}}(s)hMg(s); this is the key to bounding the number of moderate points from Section 3.
As stated at the end of Section 4.2, the value c(g)c(g)c(g)c(g)c(g) can be expressed in terms of geometry properties of the image of Sg[M+1]Sg[M+1]S_(g)^([M+1])\mathfrak{S}_{g}^{[M+1]}Sg[M+1] under the Faltings-Zhang morphism.
Question 5.2. What is an admissible value for c(g)c(g)c(g)c(g)c(g) ?
5.2. Bounding the number of moderate points-a sketch
Recall that, by the discussion at the end of Section 3, we need to bound the number of moderate points.
We retain the notation of Sections 3 and 5. The curve CCCCC from Section 3 can be equipped with suitable level structure over a field F′/FF′/FF^(')//FF^{\prime} / FF′/F with [F′:F]F′:F[F^('):F]\left[F^{\prime}: F\right][F′:F] bounded in terms of ggggg. The rank of Jac(C)(F′)Jacâ¡(C)F′Jac(C)(F^('))\operatorname{Jac}(C)\left(F^{\prime}\right)Jacâ¡(C)(F′) may be dangerously larger than the rank of Jac(C)(F)Jacâ¡(C)(F)Jac(C)(F)\operatorname{Jac}(C)(F)Jacâ¡(C)(F). But recall that we are interested in bounding #C(F)#C(F)#C(F)\# C(F)#C(F) from above, so only the groupJac(C)(F)groupâ¡Jacâ¡(C)(F)group Jac(C)(F)\operatorname{group} \operatorname{Jac}(C)(F)groupâ¡Jacâ¡(C)(F) will be relevant. Moreover, c1,c2c1,c2c_(1),c_(2)c_{1}, c_{2}c1,c2, and c3c3c_(3)c_{3}c3 from a suitable version of Vojta's inequality are unaffected by extending FFFFF. The effect is that we may identify CCCCC with a fiber of CgCgC_(g)\mathfrak{C}_{g}Cg above some point s∈Mg(F′)s∈MgF′s inM_(g)(F^('))s \in \mathbb{M}_{g}\left(F^{\prime}\right)s∈Mg(F′). For simplicity, we assume F=F′F=F′F=F^(')F=F^{\prime}F=F′ for this proof sketch.
Suppose now that (5.3) holds, so hMg(s)hMg(s)h_(M_(g))(s)h_{\mathbb{M}_{g}}(s)hMg(s) is sufficiently large in terms of ggggg. We fix an auxiliary base point P′∈C(F)P′∈C(F)P^(')in C(F)P^{\prime} \in C(F)P′∈C(F). We must bound from above the number of points in
B(R)={P∈C(F):∥P−P′∥2≤R2} with R=(c4(g)hMg(s))1/2B(R)=P∈C(F):P−P′2≤R2 with R=c4(g)hMg(s)1/2B(R)={P in C(F):||P-P^(')||^(2) <= R^(2)}quad" with "R=(c_(4)(g)h_(M_(g))(s))^(1//2)B(R)=\left\{P \in C(F):\left\|P-P^{\prime}\right\|^{2} \leq R^{2}\right\} \quad \text { with } R=\left(c_{4}(g) h_{\mathbb{M}_{g}}(s)\right)^{1 / 2}B(R)={P∈C(F):∥P−P′∥2≤R2} with R=(c4(g)hMg(s))1/2
Recall M=3g−2M=3g−2M=3g-2M=3 g-2M=3g−2 and suppose P0,…,PM∈C(F)P0,…,PM∈C(F)P_(0),dots,P_(M)in C(F)P_{0}, \ldots, P_{M} \in C(F)P0,…,PM∈C(F). If the tuple (P0,…,PM)P0,…,PM(P_(0),dots,P_(M))\left(P_{0}, \ldots, P_{M}\right)(P0,…,PM) is in general position, i.e., (P1−P0,…,PM−P0)P1−P0,…,PM−P0(P_(1)-P_(0),dots,P_(M)-P_(0))\left(P_{1}-P_{0}, \ldots, P_{M}-P_{0}\right)(P1−P0,…,PM−P0) lies in U(Q¯)U(Q¯)U( bar(Q))U(\overline{\mathbb{Q}})U(Q¯) from (5.4), then there is iiiii with
Pi∉B(P0,r)={P∈C(F):∥P−P0∥2≤r2} with r=(c5(g)hMg(s))1/2Pi∉BP0,r=P∈C(F):P−P02≤r2 with r=c5(g)hMg(s)1/2P_(i)!in B(P_(0),r)={P in C(F):||P-P_(0)||^(2) <= r^(2)}quad" with "r=(c_(5)(g)h_(M_(g))(s))^(1//2)P_{i} \notin B\left(P_{0}, r\right)=\left\{P \in C(F):\left\|P-P_{0}\right\|^{2} \leq r^{2}\right\} \quad \text { with } r=\left(c_{5}(g) h_{\mathbb{M}_{g}}(s)\right)^{1 / 2}Pi∉B(P0,r)={P∈C(F):∥P−P0∥2≤r2} with r=(c5(g)hMg(s))1/2
If we had a guarantee that such (M+1)(M+1)(M+1)(M+1)(M+1)-tuples of pairwise distinct points are always in general position, then #B(P0,r)<M=3g−2#BP0,r<M=3g−2#B(P_(0),r) < M=3g-2\# B\left(P_{0}, r\right)<M=3 g-2#B(P0,r)<M=3g−2. By sphere packing, we can cover the image of B(R)B(R)B(R)B(R)B(R) in Jac(C)(F)⊗RJacâ¡(C)(F)⊗RJac(C)(F)oxR\operatorname{Jac}(C)(F) \otimes \mathbb{R}Jacâ¡(C)(F)⊗R by at most (1+2R/r)rkJac(C)(F)(1+2R/r)rkJac(C)(F)(1+2R//r)^(rkJac(C)(F))(1+2 R / r)^{\mathrm{rkJac}(C)(F)}(1+2R/r)rkJac(C)(F) closed balls in Jac(C)(F)Jacâ¡(C)(F)Jac(C)(F)\operatorname{Jac}(C)(F)Jacâ¡(C)(F) of radius rrrrr. One can even arrange for the ball centers to arise as points of C(F)C(F)C(F)C(F)C(F). The modular height hMg(s)hMg(s)h_(M_(g))(s)h_{\mathbb{M}_{g}}(s)hMg(s) cancels out in the quotient
This would complete the proof of Theorem 1.12 except that there is no reason to believe that (P1−P0,…,PM−P0)∈U(Q¯)P1−P0,…,PM−P0∈U(Q¯)(P_(1)-P_(0),dots,P_(M)-P_(0))in U( bar(Q))\left(P_{1}-P_{0}, \ldots, P_{M}-P_{0}\right) \in U(\overline{\mathbb{Q}})(P1−P0,…,PM−P0)∈U(Q¯) (even if the PjPjP_(j)P_{j}Pj are pairwise distinct). Treating points with image in the complement of UUUUU requires induction on the dimension. Here we rely on the freedom to replace MgMgM_(g)\mathbb{M}_{g}Mg by a subvariety in that Gao's Theorem 5.1.
Let us briefly explain the resulting induction step. Observe that the dimension of this exceptional set is at most dimD(Cg[M+1])−1≤M+dimMgdimâ¡DCg[M+1]−1≤M+dimâ¡Mgdim D(C_(g)^([M+1]))-1 <= M+dim M_(g)\operatorname{dim} \mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right)-1 \leq M+\operatorname{dim} \mathbb{M}_{g}dimâ¡D(Cg[M+1])−1≤M+dimâ¡Mg. There are two cases for (P0,…,PM)P0,…,PM(P_(0),dots,P_(M))\left(P_{0}, \ldots, P_{M}\right)(P0,…,PM) with image in the exceptional set (D(Cg[M+1])∖U)(Q¯)DCg[M+1]∖U(Q¯)(D(C_(g)^([M+1]))\\U)( bar(Q))\left(\mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right) \backslash U\right)(\overline{\mathbb{Q}})(D(Cg[M+1])∖U)(Q¯) on which we do not have the height inequality. For the case study, recall that s∈Mg(Q¯)s∈Mg(Q¯)s inM_(g)( bar(Q))s \in \mathbb{M}_{g}(\overline{\mathbb{Q}})s∈Mg(Q¯) denotes the point below all the PjPjP_(j)P_{j}Pj and τ(s)∈Ag(Q¯)Ï„(s)∈Ag(Q¯)tau(s)inA_(g)( bar(Q))\tau(s) \in \mathbb{A}_{g}(\overline{\mathbb{Q}})Ï„(s)∈Ag(Q¯) is its image under the Torelli morphism τÏ„tau\tauÏ„.
First, assume that the fiber of D(Cg[M+1])∖U→AgDCg[M+1]∖U→AgD(C_(g)^([M+1]))\\U rarrAg\mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right) \backslash U \rightarrow \mathbb{A} gD(Cg[M+1])∖U→Ag above τ(s)Ï„(s)tau(s)\tau(s)Ï„(s) has dimension at most MMMMM. This fiber contains (P1−P0,…,PM−P0)P1−P0,…,PM−P0(P_(1)-P_(0),dots,P_(M)-P_(0))\left(P_{1}-P_{0}, \ldots, P_{M}-P_{0}\right)(P1−P0,…,PM−P0). This case is solved using a zero estimate motivated by the following simple lemma.
Lemma 5.3. Suppose CCCCC is an irreducible curve defined over CCC\mathbb{C}C and WWWWW a proper Zariski closed subset of CMCMC^(M)C^{M}CM. If Σ⊆C(C)Σ⊆C(C)Sigma sube C(C)\Sigma \subseteq C(\mathbb{C})Σ⊆C(C) with ΣM⊆W(C)ΣM⊆W(C)Sigma^(M)sube W(C)\Sigma^{M} \subseteq W(\mathbb{C})ΣM⊆W(C), then ΣΣSigma\SigmaΣ is finite.
The second case is if the fiber of D(Cg[M+1])∖U→AgDCg[M+1]∖U→AgD(C_(g)^([M+1]))\\U rarrA_(g)\mathscr{D}\left(\mathfrak{C}_{g}^{[M+1]}\right) \backslash U \rightarrow \mathbb{A}_{g}D(Cg[M+1])∖U→Ag above τ(s)Ï„(s)tau(s)\tau(s)Ï„(s) has dimension at least M+1M+1M+1M+1M+1. For dimension reasons, sssss lies in a proper subvariety SSSSS of MgMgM_(g)\mathbb{M}_{g}Mg. Here we apply induction on the dimension and replace MgMgM_(g)\mathbb{M}_{g}Mg by its subvariety SSSSS.
This completes the proof sketch.
Kühne [39] combined ideas from equidistribution with the approach laid out in [24] to get a suitable uniform estimate for #B(P0,r)#BP0,r#B(P_(0),r)\# B\left(P_{0}, r\right)#B(P0,r) without the restriction (5.3) on hMg(s)hMg(s)h_(M_(g))(s)h_{\mathbb{M}_{g}}(s)hMg(s). Yuan's Theorem 1.1 [66] does so as well, but he follows a different approach. He obtains a more general estimate that works also in the function field setting and allows for a larger RRRRR.
6. HYPERELLIPTIC CURVES
A hyperelliptic curve is a smooth curve of genus at least 2 that admits a degree 2 morphism to the projective line. Hyperelliptic curves have particularly simple planar models. Indeed, if the base field is a number field FFFFF, then a hyperelliptic curve of genus ggggg can be
represented by a hyperelliptic equation
Y2=f(X)Y2=f(X)Y^(2)=f(X)quadY^{2}=f(X) \quadY2=f(X) with f∈F[X]f∈F[X]f in F[X]f \in F[X]f∈F[X] monic and square-free of degree 2g+12g+12g+12 g+12g+1 or 2g+22g+22g+22 g+22g+2.
In this section we determine consequences of Theorem 1.12 for hyperelliptic curves. Our aim is to leave the world of curves and Jacobians and to present a bound for the number of rational solutions of Y2=f(X)Y2=f(X)Y^(2)=f(X)Y^{2}=f(X)Y2=f(X) that can be expressed in terms of fffff. We refer to Section 6 of [23] for a similar example in a 1-parameter family of hyperelliptic curves.
To keep technicalities to a minimum, we assume that our base field is F=QF=QF=QF=\mathbb{Q}F=Q and that f∈Z[X]f∈Z[X]f inZ[X]f \in \mathbb{Z}[X]f∈Z[X] is monic of degree d=2g+1d=2g+1d=2g+1d=2 g+1d=2g+1 and factors into linear factors in Q[X]Q[X]Q[X]\mathbb{Q}[X]Q[X]. The curve represented by the hyperelliptic equation has a marked Weierstrass point "at infinity." These assumptions can be loosened with some extra effort. For example, if fffff does not factor in Q[X]Q[X]Q[X]\mathbb{Q}[X]Q[X], then the class number of the splitting field will play a part.
Say, f=Xd+fd−1Xd−1+⋯+f0f=Xd+fd−1Xd−1+⋯+f0f=X^(d)+f_(d-1)X^(d-1)+cdots+f_(0)f=X^{d}+f_{d-1} X^{d-1}+\cdots+f_{0}f=Xd+fd−1Xd−1+⋯+f0. By the assumption above, f=(X−α1)⋯(X−αd)f=X−α1⋯X−αdf=(X-alpha_(1))cdots(X-alpha_(d))f=\left(X-\alpha_{1}\right) \cdots\left(X-\alpha_{d}\right)f=(X−α1)⋯(X−αd) with α1,…,αd∈Qα1,…,αd∈Qalpha_(1),dots,alpha_(d)inQ\alpha_{1}, \ldots, \alpha_{d} \in \mathbb{Q}α1,…,αd∈Q which are necessarily integers. The discriminant of fffff is
We have the following estimate for the cardinality. Below ω(n)ω(n)omega(n)\omega(n)ω(n) denotes the number of distinct prime divisors of n∈Z∖{0}n∈Z∖{0}n inZ\\{0}n \in \mathbb{Z} \backslash\{0\}n∈Z∖{0}.
Theorem 6.1. Let g≥2g≥2g >= 2g \geq 2g≥2. There exist c(g)>1c(g)>1c(g) > 1c(g)>1c(g)>1 and c′(g)>0c′(g)>0c^(')(g) > 0c^{\prime}(g)>0c′(g)>0 with the following property. Suppose f∈Z[X]f∈Z[X]f inZ[X]f \in \mathbb{Z}[X]f∈Z[X] is monic of degree 2g+12g+12g+12 g+12g+1, square-free, and factors into linear factors in Q[X]Q[X]Q[X]\mathbb{Q}[X]Q[X]. Then
Proof. The hyperelliptic equation represents a curve CCCCC defined over QQQ\mathbb{Q}Q of genus ggggg.
If ppppp is a prime number with p∤Δfp∤Δfp∤Delta_(f)p \nmid \Delta_{f}p∤Δf, then the αiαialpha_(i)\alpha_{i}αi are pairwise distinct modulo ppppp. If ppppp is also odd, then the equation Y2=f(X)Y2=f(X)Y^(2)=f(X)Y^{2}=f(X)Y2=f(X) reduced modulo ppppp defines a hyperelliptic curve over FpFpF_(p)\mathbb{F}_{p}Fp. So CCCCC has good reduction at all primes that do not divide 2Δf2Δf2Delta_(f)2 \Delta_{f}2Δf. Thus the Jacobian Jac(C)Jacâ¡(C)Jac(C)\operatorname{Jac}(C)Jacâ¡(C) has good reduction at the same primes.
We may embed CCCCC into its Jacobian Jac(C)Jacâ¡(C)Jac(C)\operatorname{Jac}(C)Jacâ¡(C) by sending the marked Weierstrass point to 0 . Each root αiαialpha_(i)\alpha_{i}αi of fffff corresponds to a rational point in C(Q)C(Q)C(Q)C(\mathbb{Q})C(Q) and it is sent to a point of order 2 in Jac(C)Jacâ¡(C)Jac(C)\operatorname{Jac}(C)Jacâ¡(C). Moreover, these points generate the 2-torsion in Jac(C)tors Jacâ¡(C)tors Jac(C)_("tors ")\operatorname{Jac}(C)_{\text {tors }}Jacâ¡(C)tors . In particular, all points of order 2 in Jac(C)tors Jacâ¡(C)tors Jac(C)_("tors ")\operatorname{Jac}(C)_{\text {tors }}Jacâ¡(C)tors are rational.
Next we bound the rank of Jac(C)(Q)Jacâ¡(C)(Q)Jac(C)(Q)\operatorname{Jac}(C)(\mathbb{Q})Jacâ¡(C)(Q) from above. Indeed, we could use the work of Ooe-Top [49] or [37, THEOREM c.1.9]. The latter applied to Jac(C),k=Q,m=2Jacâ¡(C),k=Q,m=2Jac(C),k=Q,m=2\operatorname{Jac}(C), k=\mathbb{Q}, m=2Jacâ¡(C),k=Q,m=2, and SSSSS the prime divisors of 2Δf2Δf2Delta_(f)2 \Delta_{f}2Δf yields rkJac(C)(Q)≤2g#S≤2gω(2Δf)≤2g+2gω(Δf)rkâ¡Jacâ¡(C)(Q)≤2g#S≤2gω2Δf≤2g+2gωΔfrk Jac(C)(Q) <= 2g#S <= 2g omega(2Delta_(f)) <= 2g+2g omega(Delta_(f))\operatorname{rk} \operatorname{Jac}(C)(\mathbb{Q}) \leq 2 g \# S \leq 2 g \omega\left(2 \Delta_{f}\right) \leq 2 g+2 g \omega\left(\Delta_{f}\right)rkâ¡Jacâ¡(C)(Q)≤2g#S≤2gω(2Δf)≤2g+2gω(Δf) Here we use that QQQ\mathbb{Q}Q has a trivial class group; in a more general setup, the class group of the
splitting field of fffff will enter at this point. The estimate (6.1) follows from Theorem 1.12 in the case F=QF=QF=QF=\mathbb{Q}F=Q with adjusted constants.
It is tempting to average (6.1) over the fffff bounded in a suitable way, e.g., by bounding the maximal modulus of the roots by a parameter XXXXX. As pointed out to the author by Christian Elsholtz and Martin Widmer, this average will be unbounded as X→∞X→∞X rarr ooX \rightarrow \inftyX→∞.
ACKNOWLEDGMENTS
Much of the work presented here was obtained together with my coauthors. In particular, I thank Serge Cantat, Vesselin Dimitrov, Ziyang Gao, and Junyi Xie for the fruitful collaboration. I also thank Gabriel Dill, Ziyang Gao, Lars Kühne, and David Masser for comments on this survey.
FUNDING
The author has received funding from the Swiss National Science Foundation project n๠200020_184623.
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Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland, philipp.habegger@unibas.ch
THETA LIFTING AND LANGLANDS FUNCTORIALITY
ATSUSHI ICHINO
ABSTRACT
We review various aspects of theta lifting and its role in studying Langlands functoriality. In particular, we discuss realizations of the Jacquet-Langlands correspondence and the Shimura-Waldspurger correspondence in terms of theta lifting and their arithmetic applications.
Langlands functoriality is a principle relating two different kinds of automorphic forms and plays a pivotal role in number theory. Before Langlands formulated this principle in [42], this phenomenon was already observed in the following classical example discovered by Eichler [15] and developed by Shimizu [48]. Consider the space
of elliptic cusp forms of weight kkkkk and level NNNNN, where kkkkk and NNNNN are positive integers and Γ0(N)Γ0(N)Gamma_(0)(N)\Gamma_{0}(N)Γ0(N) is the congruence subgroup given by
Γ0(N)={(abcd)∈SL2(Z)|c≡0modN}Γ0(N)=abcd∈SL2(Z)c≡0modNGamma_(0)(N)={([a,b],[c,d])inSL_(2)(Z)|c-=0mod N}\Gamma_{0}(N)=\left\{\left.\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right) \in \mathrm{SL}_{2}(\mathbb{Z}) \right\rvert\, c \equiv 0 \bmod N\right\}Γ0(N)={(abcd)∈SL2(Z)|c≡0modN}
This space consists of holomorphic functions fffff on the upper half-plane SSS\mathfrak{S}S which satisfy
for all γ∈Γ0(N)γ∈Γ0(N)gamma inGamma_(0)(N)\gamma \in \Gamma_{0}(N)γ∈Γ0(N) and z∈Sz∈Sz inSz \in \mathfrak{S}z∈S and which vanish at all cusps. Here SL2(R)SL2(R)SL_(2)(R)\mathrm{SL}_{2}(\mathbb{R})SL2(R) acts on SSS\mathscr{S}S by linear fractional transformations and j((abcd),z)=cz+djabcd,z=cz+dj(([a,b],[c,d]),z)=cz+dj\left(\left(\begin{array}{ll}a & b \\ c & d\end{array}\right), z\right)=c z+dj((abcd),z)=cz+d is the factor of automorphy. It is also equipped with the action of Hecke operators TnTnT_(n)T_{n}Tn for all positive integers nnnnn, which is a central tool in the arithmetic study of automorphic forms. On the other hand, to every indefinite quaternion division algebra BBBBB over QQQ\mathbb{Q}Q, we may associate a space
of modular forms, where ΓBΓBGamma_(B)\Gamma_{B}ΓB is the group of norm-one elements in BBBBB. Namely, this space is defined similarly by replacing Γ0(N)Γ0(N)Gamma_(0)(N)\Gamma_{0}(N)Γ0(N) by ΓBΓBGamma_(B)\Gamma_{B}ΓB (which can be regarded as a subgroup of (B⊗QR)×≅GL2(R))B⊗QR×≅GL2(R){:(Box_(Q)R)^(xx)~=GL_(2)(R))\left.\left(B \otimes_{\mathbb{Q}} \mathbb{R}\right)^{\times} \cong \mathrm{GL}_{2}(\mathbb{R})\right)(B⊗QR)×≅GL2(R)) and is equipped with the action of Hecke operators TnBTnBT_(n)^(B)T_{n}^{B}TnB. Now assume that NNNNN is the product of an even number of distinct primes and BBBBB is ramified precisely at the primes dividing NNNNN. Then by the works of Eichler and Shimizu, the trace of TnBTnBT_(n)^(B)T_{n}^{B}TnB on Sk(ΓB)SkΓBS_(k)(Gamma_(B))S_{k}\left(\Gamma_{B}\right)Sk(ΓB) coincides with the trace of TnTnT_(n)T_{n}Tn on the new part of Sk(Γ0(N))SkΓ0(N)S_(k)(Gamma_(0)(N))S_{k}\left(\Gamma_{0}(N)\right)Sk(Γ0(N)) for all nnnnn prime to NNNNN.
This remarkable relation was thoroughly studied by Jacquet-Langlands [37] in the framework of automorphic representations. Let FFFFF be a number field with adèle ring AAA\mathbb{A}A. Let BBBBB be a quaternion division algebra over FFFFF. Then Jacquet-Langlands proved that for any irreducible automorphic representation πB≅⊗vπvBÏ€B≅⊗vÏ€vBpi^(B)~=ox_(v)pi_(v)^(B)\pi^{B} \cong \otimes_{v} \pi_{v}^{B}Ï€B≅⊗vÏ€vB of B×(A)B×(A)B^(xx)(A)B^{\times}(\mathbb{A})B×(A), there exists a unique irreducible automorphic representation π≅⊗vπvπ≅⊗vÏ€vpi~=ox_(v)pi_(v)\pi \cong \otimes_{v} \pi_{v}π≅⊗vÏ€v of GL2(A)GL2(A)GL_(2)(A)\mathrm{GL}_{2}(\mathbb{A})GL2(A) such that
for almost all places vvvvv of FFFFF. Moreover, they described the image of this map πB↦πÏ€B↦πpi^(B)|->pi\pi^{B} \mapsto \piÏ€B↦π precisely.
The Jacquet-Langlands correspondence gives a basic example of Langlands functoriality. To explain this, let GGGGG be a connected reductive group over FFFFF. Let LGLG^(L)G{ }^{L} GLG be the LLLLL-group of GGGGG, which was introduced by Langlands and which should govern automorphic representations of G(A)G(A)G(A)G(\mathbb{A})G(A). Explicitly, LGLG^(L)G{ }^{L} GLG is defined as a semiproduct G^⋊ΓFG^⋊ΓFhat(G)><|Gamma_(F)\hat{G} \rtimes \Gamma_{F}G^⋊ΓF, where G^G^hat(G)\hat{G}G^ is the complex dual group of G,ΓF=Gal(F¯/F)G,ΓF=Galâ¡(F¯/F)G,Gamma_(F)=Gal( bar(F)//F)G, \Gamma_{F}=\operatorname{Gal}(\bar{F} / F)G,ΓF=Galâ¡(F¯/F) is the absolute Galois group of FFFFF, and the action of ΓFΓFGamma_(F)\Gamma_{F}ΓF on G^G^hat(G)\hat{G}G^ is inherited from the action of ΓFΓFGamma_(F)\Gamma_{F}ΓF on the root datum of GGGGG. To motivate it, let us admit for the moment the existence of the hypothetical Langlands group LFLFL_(F)\mathscr{L}_{F}LF over FFFFF, which is equipped with a surjection LF→ΓFLF→ΓFL_(F)rarrGamma_(F)\mathscr{L}_{F} \rightarrow \Gamma_{F}LF→ΓF. Then it is conjectured that irreducible automorphic representations of G(A)G(A)G(A)G(\mathbb{A})G(A) are classified in terms of certain LLLLL-homomorphisms LF→LGLF→LGL_(F)rarr^(L)G\mathscr{L}_{F} \rightarrow{ }^{L} GLF→LG, i.e., homomorphisms commuting with the projections to ΓFΓFGamma_(F)\Gamma_{F}ΓF. (Strictly speaking, we consider here packets of tempered automorphic representations.) Now suppose that we have another connected reductive quasisplit group G′G′G^(')G^{\prime}G′ over FFFFF and an LLLLL-homomorphism
r:LG→LG′r:LG→LG′r:^(L)G rarr^(L)G^(')r:{ }^{L} G \rightarrow{ }^{L} G^{\prime}r:LG→LG′
Let πÏ€pi\piÏ€ be an irreducible automorphic representation of G(A)G(A)G(A)G(\mathbb{A})G(A) which should correspond to an LLLLL-homomorphism
Then Langlands functoriality predicts the existence of an irreducible automorphic representation π′π′pi^(')\pi^{\prime}π′ of G′(A)G′(A)G^(')(A)G^{\prime}(\mathbb{A})G′(A) which should correspond to the LLLLL-homomorphism
This conjectural relation between πÏ€pi\piÏ€ and π′π′pi^(')\pi^{\prime}π′ can be formulated without assuming the existence of LFLFL_(F)\mathscr{L}_{F}LF as follows. Recall that for almost all places vvvvv of FFFFF, the local component πvÏ€vpi_(v)\pi_{v}Ï€v of πÏ€pi\piÏ€ at vvvvv is unramified, so that it determines and is determined by a G^G^hat(G)\hat{G}G^-conjugacy class c(πv)cÏ€vc(pi_(v))c\left(\pi_{v}\right)c(Ï€v) in LGLG^(L)G{ }^{L} GLG via the Satake isomorphism. Then π′π′pi^(')\pi^{\prime}π′ should satisfy
for almost all vvvvv. Note that the Jacquet-Langlands correspondence mentioned above is the special case when G=B×,G′=GL2G=B×,G′=GL2G=B^(xx),G^(')=GL_(2)G=B^{\times}, G^{\prime}=\mathrm{GL}_{2}G=B×,G′=GL2 (so that LG=LG′=GL2(C)×ΓFLG=LG′=GL2(C)×ΓF^(L)G=^(L)G^(')=GL_(2)(C)xxGamma_(F){ }^{L} G={ }^{L} G^{\prime}=\mathrm{GL}_{2}(\mathbb{C}) \times \Gamma_{F}LG=LG′=GL2(C)×ΓF ), and rrrrr is the identity map.
Although Langlands functoriality is out of reach in general, it led to substantial developments in the theory of automorphic forms. For example, the trace formula was developed by Arthur to study automorphic representations, culminating in his book [1] which establishes the case when GGGGG is a symplectic group or a quasisplit special orthogonal group, G′G′G^(')G^{\prime}G′ is a general linear group, and rrrrr is the standard embedding. There are also other methods to attack Langlands functoriality, such as the converse theorem [12,13], the automorphic descent [24], and the theta lifting. In this report, we will discuss various aspects of the theta lifting, which can be viewed as an explicit realization in the case when (G,G′)G,G′(G,G^('))\left(G, G^{\prime}\right)(G,G′) is a certain pair of classical groups.
2. THETA LIFTING
In this section, we recall the notion of the theta lifting, with emphasis on the realization of the Jacquet-Langlands correspondence. We also review some of its applications to explicit formulas for automorphic periods in terms of special values of LLLLL-functions.
2.1. Basic definitions and properties
Let FFFFF be a number field with adèle ring A=AFA=AFA=A_(F)\mathbb{A}=\mathbb{A}_{F}A=AF. Let WWWWW be a symplectic space over FFFFF equipped with a nondegenerate bilinear alternating form (⋅,⋅)W(â‹…,â‹…)W(*,*)_(W)(\cdot, \cdot)_{W}(â‹…,â‹…)W and let Sp(W)Spâ¡(W)Sp(W)\operatorname{Sp}(W)Spâ¡(W) denote the symplectic group of WWWWW. Similarly, let VVVVV be a quadratic space over FFFFF equipped with a nondegenerate bilinear symmetric form (⋅,⋅)V(â‹…,â‹…)V(*,*)_(V)(\cdot, \cdot)_{V}(â‹…,â‹…)V and let O(V)O(V)O(V)\mathrm{O}(V)O(V) denote the orthogonal group of VVVVV. Then the pair
is an example of a reductive dual pair introduced by Howe [30]. Namely, if we consider the symplectic space W=W⊗FVW=W⊗FVW=Wox_(F)V\mathbb{W}=W \otimes_{F} VW=W⊗FV equipped with the form (⋅,⋅)W⊗(⋅,⋅)V(â‹…,â‹…)W⊗(â‹…,â‹…)V(*,*)_(W)ox(*,*)_(V)(\cdot, \cdot)_{W} \otimes(\cdot, \cdot)_{V}(â‹…,â‹…)W⊗(â‹…,â‹…)V and the natural homomorphism
then Sp(W)Sp(W)Sp(W)\mathrm{Sp}(W)Sp(W) and O(V)O(V)O(V)\mathrm{O}(V)O(V) are mutual commutants in Sp(W)Sp(W)Sp(W)\mathrm{Sp}(\mathbb{W})Sp(W).
Roughly speaking, the theta lifting is an integral transform with kernel given by a particular automorphic form on Sp(W)(A)Spâ¡(W)(A)Sp(W)(A)\operatorname{Sp}(\mathbb{W})(\mathbb{A})Spâ¡(W)(A) restricted to Sp(W)(A)×O(V)(A)Spâ¡(W)(A)×O(V)(A)Sp(W)(A)xxO(V)(A)\operatorname{Sp}(W)(\mathbb{A}) \times \mathrm{O}(V)(\mathbb{A})Spâ¡(W)(A)×O(V)(A). To be precise, we need to consider the metaplectic group Mp(W)(A)Mp(W)(A)Mp(W)(A)\mathrm{Mp}(\mathbb{W})(\mathbb{A})Mp(W)(A), which is a nontrivial topological central extension
(Here we have abused notation since Mp(W)(A)Mpâ¡(W)(A)Mp(W)(A)\operatorname{Mp}(\mathbb{W})(\mathbb{A})Mpâ¡(W)(A) is not the group of AAA\mathbb{A}A-valued points of an algebraic group over FFFFF.) This extension splits over Sp(W)(F)Spâ¡(W)(F)Sp(W)(F)\operatorname{Sp}(\mathbb{W})(F)Spâ¡(W)(F) canonically, so that we may speak of automorphic forms on Mp(W)(A)Mpâ¡(W)(A)Mp(W)(A)\operatorname{Mp}(\mathbb{W})(\mathbb{A})Mpâ¡(W)(A). We are interested in a particular representation ωωomega\omegaω of Mp(W)(A)Mpâ¡(W)(A)Mp(W)(A)\operatorname{Mp}(\mathbb{W})(\mathbb{A})Mpâ¡(W)(A) (depending on a choice of a nontrivial additive character of A/FA/FA//F\mathbb{A} / FA/F ), called the Weil representation [61], which is a representation theoretic incarnation of theta functions. This representation has an automorphic realization, i.e., there is an Mp(W)(A)Mpâ¡(W)(A)Mp(W)(A)\operatorname{Mp}(\mathbb{W})(\mathbb{A})Mpâ¡(W)(A)-equivariant mapφ↦θφmapâ¡Ï†â†¦Î¸Ï†map varphi|->theta_(varphi)\operatorname{map} \varphi \mapsto \theta_{\varphi}mapâ¡Ï†â†¦Î¸Ï† from ωωomega\omegaω to the space of automorphic forms on Mp(W)(A)Mpâ¡(W)(A)Mp(W)(A)\operatorname{Mp}(\mathbb{W})(\mathbb{A})Mpâ¡(W)(A). On the other hand, there exists a dotted arrow making the following diagram commute:
(Note that it descends to a homomorphism from the bottom left corner if and only if dimVdimâ¡Vdim V\operatorname{dim} Vdimâ¡V is even.) Thus we may regard θφθφtheta_(varphi)\theta_{\varphi}θφ as an automorphic form on Mp(W)(A)×O(V)(A)Mpâ¡(W)(A)×O(V)(A)Mp(W)(A)xxO(V)(A)\operatorname{Mp}(W)(\mathbb{A}) \times \mathrm{O}(V)(\mathbb{A})Mpâ¡(W)(A)×O(V)(A) by restriction and associate to an automorphic form fffff on Mp(W)(A)Mpâ¡(W)(A)Mp(W)(A)\operatorname{Mp}(W)(\mathbb{A})Mpâ¡(W)(A) an automorphic form θφ(f)θφ(f)theta_(varphi)(f)\theta_{\varphi}(f)θφ(f) on O(V)(A)O(V)(A)O(V)(A)\mathrm{O}(V)(\mathbb{A})O(V)(A) by setting
θφ(f)(h)=∫Sp(W)(F)∖Mp(W)(A)θφ(g,h)f(g)¯dgθφ(f)(h)=∫Spâ¡(W)(F)∖Mpâ¡(W)(A) θφ(g,h)f(g)¯dgtheta_(varphi)(f)(h)=int_(Sp(W)(F)\\Mp(W)(A))theta_(varphi)(g,h) bar(f(g))dg\theta_{\varphi}(f)(h)=\int_{\operatorname{Sp}(W)(F) \backslash \operatorname{Mp}(W)(\mathbb{A})} \theta_{\varphi}(g, h) \overline{f(g)} d gθφ(f)(h)=∫Spâ¡(W)(F)∖Mpâ¡(W)(A)θφ(g,h)f(g)¯dg
provided the integral converges, e.g., if fffff is cuspidal.
For any irreducible cuspidal automorphic representation πÏ€pi\piÏ€ of Mp(W)(A)Mpâ¡(W)(A)Mp(W)(A)\operatorname{Mp}(W)(\mathbb{A})Mpâ¡(W)(A), we define the theta lift θ(π)θ(Ï€)theta(pi)\theta(\pi)θ(Ï€) of πÏ€pi\piÏ€ as the automorphic representation of O(V)(A)O(V)(A)O(V)(A)O(V)(\mathbb{A})O(V)(A) spanned by θφ(f)θφ(f)theta_(varphi)(f)\theta_{\varphi}(f)θφ(f) for all φ∈ωφ∈ωvarphi in omega\varphi \in \omegaφ∈ω and f∈πf∈πf in pif \in \pif∈π. We only consider the case when πÏ€pi\piÏ€ descends (resp. does not descend)
to a representation of Sp(W)(A)Spâ¡(W)(A)Sp(W)(A)\operatorname{Sp}(W)(\mathbb{A})Spâ¡(W)(A) if dimVdimâ¡Vdim V\operatorname{dim} Vdimâ¡V is even (resp. odd); otherwise θ(π)θ(Ï€)theta(pi)\theta(\pi)θ(Ï€) is always zero. To describe θ(π)θ(Ï€)theta(pi)\theta(\pi)θ(Ï€), we need to introduce the local analog of the theta lifting. First, note that the map (φ,f)↦θφ(f)(φ,f)↦θφ(f)(varphi,f)|->theta_(varphi)(f)(\varphi, f) \mapsto \theta_{\varphi}(f)(φ,f)↦θφ(f) defines an element in
since πÏ€pi\piÏ€ is unitary. Recall that ωωomega\omegaω can be regarded as the restricted tensor product of the local Weil representations ωvωvomega_(v)\omega_{v}ωv of Mp(W)(Fv)Mpâ¡(W)FvMp(W)(F_(v))\operatorname{Mp}(\mathbb{W})\left(F_{v}\right)Mpâ¡(W)(Fv) via the surjection ∏v′Mp(W)(Fv)→Mp(W)(A)âˆv′ Mpâ¡(W)Fv→Mpâ¡(W)(A)prod_(v)^(')Mp(W)(F_(v))rarr Mp(W)(A)\prod_{v}^{\prime} \operatorname{Mp}(\mathbb{W})\left(F_{v}\right) \rightarrow \operatorname{Mp}(\mathbb{W})(\mathbb{A})âˆv′Mpâ¡(W)(Fv)→Mpâ¡(W)(A), where Mp(W)(Fv)Mpâ¡(W)FvMp(W)(F_(v))\operatorname{Mp}(\mathbb{W})\left(F_{v}\right)Mpâ¡(W)(Fv) is the metaplectic cover of Sp(W)(Fv)Spâ¡(W)FvSp(W)(F_(v))\operatorname{Sp}(\mathbb{W})\left(F_{v}\right)Spâ¡(W)(Fv). Similarly, πÏ€pi\piÏ€ can be decomposed as π≅⊗vπvπ≅⊗vÏ€vpi~=ox_(v)pi_(v)\pi \cong \otimes_{v} \pi_{v}π≅⊗vÏ€v, where πvÏ€vpi_(v)\pi_{v}Ï€v is an irreducible representation of Mp(W)(Fv)Mpâ¡(W)FvMp(W)(F_(v))\operatorname{Mp}(W)\left(F_{v}\right)Mpâ¡(W)(Fv). We define the local theta lift θ(πv)θπvtheta(pi_(v))\theta\left(\pi_{v}\right)θ(Ï€v) of πvÏ€vpi_(v)\pi_{v}Ï€v as an irreducible representation of O(V)(Fv)O(V)FvO(V)(F_(v))\mathrm{O}(V)\left(F_{v}\right)O(V)(Fv) such that
which is unique (if it exists) by the Howe duality [23,31,55][23,31,55][23,31,55][23,31,55][23,31,55]. (When such a representation does not exist, we interpret θ(πv)θπvtheta(pi_(v))\theta\left(\pi_{v}\right)θ(Ï€v) as zero.) Now assume that θ(π)θ(Ï€)theta(pi)\theta(\pi)θ(Ï€) is nonzero and cuspidal. Then it follows from the Howe duality that θ(π)θ(Ï€)theta(pi)\theta(\pi)θ(Ï€) is irreducible and can be decomposed as
Remark 2.1. We may extend the Weil representation and define the theta lifting for the pair (GSp(W),GO(V))(GSpâ¡(W),GOâ¡(V))(GSp(W),GO(V))(\operatorname{GSp}(W), \operatorname{GO}(V))(GSpâ¡(W),GOâ¡(V)), where GSp(W)GSpâ¡(W)GSp(W)\operatorname{GSp}(W)GSpâ¡(W) and GO(V)GOâ¡(V)GO(V)\operatorname{GO}(V)GOâ¡(V) are the similitude groups of WWWWW and VVVVV, respectively.
2.2. Explicit realization of the Jacquet-Langlands correspondence
From now on, we mainly consider the case when
dimW=2,dimV=4dimâ¡W=2,dimâ¡V=4dim W=2,quad dim V=4\operatorname{dim} W=2, \quad \operatorname{dim} V=4dimâ¡W=2,dimâ¡V=4
and the discriminant of VVVVV is trivial. Then we may identify WWWWW with the space F2F2F^(2)F^{2}F2, equipped with the form ((x1,x2),(y1,y2))W=x1y2−x2y1x1,x2,y1,y2W=x1y2−x2y1((x_(1),x_(2)),(y_(1),y_(2)))_(W)=x_(1)y_(2)-x_(2)y_(1)\left(\left(x_{1}, x_{2}\right),\left(y_{1}, y_{2}\right)\right)_{W}=x_{1} y_{2}-x_{2} y_{1}((x1,x2),(y1,y2))W=x1y2−x2y1, so that
We may also identify VVVVV with a quaternion algebra BBBBB over FFFFF equipped with the form (x,y)V=TrB/F(xy∗)(x,y)V=TrB/Fâ¡xy∗(x,y)_(V)=Tr_(B//F)(xy^(**))(x, y)_{V}=\operatorname{Tr}_{B / F}\left(x y^{*}\right)(x,y)V=TrB/Fâ¡(xy∗), where TrB/FTrB/FTr_(B//F)\operatorname{Tr}_{B / F}TrB/F is the reduced trace and ∗∗***∗ is the main involution, so that
Here GO(V)0GOâ¡(V)0GO(V)^(0)\operatorname{GO}(V)^{0}GOâ¡(V)0 is the identity component of GO(V)GOâ¡(V)GO(V)\operatorname{GO}(V)GOâ¡(V) and B××B×B××B×B^(xx)xxB^(xx)B^{\times} \times B^{\times}B××B×acts on VVVVV by left and right multiplication.
Let πÏ€pi\piÏ€ be an irreducible cuspidal automorphic representation of GL2(A)GL2(A)GL_(2)(A)\mathrm{GL}_{2}(\mathbb{A})GL2(A). We regard the theta lift θ(π)θ(Ï€)theta(pi)\theta(\pi)θ(Ï€) of πÏ€pi\piÏ€ (restricted to GO(V)0(A)GOâ¡(V)0(A)GO(V)^(0)(A)\operatorname{GO}(V)^{0}(\mathbb{A})GOâ¡(V)0(A) ) as an automorphic representation of B×(A)×B×(A)B×(A)×B×(A)B^(xx)(A)xxB^(xx)(A)B^{\times}(\mathbb{A}) \times B^{\times}(\mathbb{A})B×(A)×B×(A). Then Shimizu [49] proved that
where πBÏ€Bpi^(B)\pi^{B}Ï€B is the Jacquet-Langlands transfer of πÏ€pi\piÏ€ to B×(A)B×(A)B^(xx)(A)B^{\times}(\mathbb{A})B×(A). (When πÏ€pi\piÏ€ does not transfer to B×(A)B×(A)B^(xx)(A)B^{\times}(\mathbb{A})B×(A), we interpret πBÏ€Bpi^(B)\pi^{B}Ï€B as zero.)
Remark 2.2. In [36], we gave the following variant of the above realization. Let B,B1,B2B,B1,B2B,B_(1),B_(2)B, B_{1}, B_{2}B,B1,B2 be three quaternion division algebras over FFFFF such that B=B1⋅B2B=B1â‹…B2B=B_(1)*B_(2)B=B_{1} \cdot B_{2}B=B1â‹…B2 in the Brauer group. We consider a 1-dimensional Hermitian space WWWWW over BBBBB and a 2-dimensional skew-Hermitian space VVVVV over BBBBB such that
where GU(W)GU(W)GU(W)\mathrm{GU}(W)GU(W) and GU(V)GU(V)GU(V)\mathrm{GU}(V)GU(V) are the unitary similitude groups of WWWWW and VVVVV, respectively. Let πBÏ€Bpi^(B)\pi^{B}Ï€B be an irreducible automorphic representation of B×(A)B×(A)B^(xx)(A)B^{\times}(\mathbb{A})B×(A) such that its Jacquet-Langlands transfer to GL2(A)GL2(A)GL_(2)(A)\mathrm{GL}_{2}(\mathbb{A})GL2(A) is cuspidal. Then we have
where πB1Ï€B1pi^(B_(1))\pi^{B_{1}}Ï€B1 and πB2Ï€B2pi^(B_(2))\pi^{B_{2}}Ï€B2 are the Jacquet-Langlands transfers of πBÏ€Bpi^(B)\pi^{B}Ï€B to B1×(A)B1×(A)B_(1)^(xx)(A)B_{1}^{\times}(\mathbb{A})B1×(A) and B2×(A)B2×(A)B_(2)^(xx)(A)B_{2}^{\times}(\mathbb{A})B2×(A), respectively. We believe that this realization is useful to study integral period relations.
2.3. Seesaw identities
One of the advantages of the theta lifting is that it produces various period relations in a simple way, which was observed by Kudla [39]. Suppose that we have two reductive dual pairs (G,H)(G,H)(G,H)(G, H)(G,H) and (G′,H′)G′,H′(G^('),H^('))\left(G^{\prime}, H^{\prime}\right)(G′,H′) in the same symplectic group such that G⊂G′G⊂G′G subG^(')G \subset G^{\prime}G⊂G′ and H⊃H′H⊃H′H supH^(')H \supset H^{\prime}H⊃H′. This can be illustrated by the following picture, called a seesaw diagram:
Let fffff and f′f′f^(')f^{\prime}f′ be automorphic forms on G(A)G(A)G(A)G(\mathbb{A})G(A) and H′(A)H′(A)H^(')(A)H^{\prime}(\mathbb{A})H′(A), respectively. Then the theta lifting produces automorphic forms θφ(f)θφ(f)theta_(varphi)(f)\theta_{\varphi}(f)θφ(f) and θφ(f′)θφf′theta_(varphi)(f^('))\theta_{\varphi}\left(f^{\prime}\right)θφ(f′) on H(A)H(A)H(A)H(\mathbb{A})H(A) and G′(A)G′(A)G^(')(A)G^{\prime}(\mathbb{A})G′(A), respectively, and the so-called seesaw identity
As an example of this identity, we recall Waldspurger's formula for torus periods. We keep the setup of the previous subsection. Fix a quadratic extension EEEEE of FFFFF which embeds into BBBBB and write B=E⊕EjB=E⊕EjB=E o+EjB=E \oplus E jB=E⊕Ej with a trace zero element jjjjj in BBBBB. Let V=V1⊕V2V=V1⊕V2V=V_(1)o+V_(2)V=V_{1} \oplus V_{2}V=V1⊕V2 be the corresponding decomposition of quadratic spaces, so that
Then the identification W⊗FV=(W⊗FV1)⊕(W⊗FV2)W⊗FV=W⊗FV1⊕W⊗FV2Wox_(F)V=(Wox_(F)V_(1))o+(Wox_(F)V_(2))W \otimes_{F} V=\left(W \otimes_{F} V_{1}\right) \oplus\left(W \otimes_{F} V_{2}\right)W⊗FV=(W⊗FV1)⊕(W⊗FV2) gives rise to the following seesaw diagram:
Let πBÏ€Bpi^(B)\pi^{B}Ï€B be an irreducible automorphic representation of B×(A)B×(A)B^(xx)(A)B^{\times}(\mathbb{A})B×(A) such that its JacquetLanglands transfer πÏ€pi\piÏ€ to GL2(A)GL2(A)GL_(2)(A)\mathrm{GL}_{2}(\mathbb{A})GL2(A) is cuspidal. Let χχchi\chiχ be an automorphic character of AE×AE×A_(E)^(xx)\mathbb{A}_{E}^{\times}AE×. Assume that the product of the central character of πBÏ€Bpi^(B)\pi^{B}Ï€B and the restriction of χχchi\chiχ to AF×AF×A_(F)^(xx)\mathbb{A}_{F}^{\times}AF×is trivial and consider the torus period
P(f,χ)=∫E×AF×∖AE×f(h)χ(h)dhP(f,χ)=∫E×AF×∖AE× f(h)χ(h)dhP(f,chi)=int_(E^(xx)A_(F)^(xx)\\A_(E)^(xx))f(h)chi(h)dhP(f, \chi)=\int_{E^{\times} \mathbb{A}_{F}^{\times} \backslash \mathbb{A}_{E}^{\times}} f(h) \chi(h) d hP(f,χ)=∫E×AF×∖AE×f(h)χ(h)dh
for a decomposable vector f∈πBf∈πBf inpi^(B)f \in \pi^{B}f∈πB. Then using the above seesaw diagram, Waldspurger [54] proved that
ζ(s)ζ(s)zeta(s)\zeta(s)ζ(s) is the completed Dedekind zeta function of FFFFF,
L(s,πE×χ)Ls,Ï€E×χL(s,pi_(E)xx chi)L\left(s, \pi_{E} \times \chi\right)L(s,Ï€E×χ) is the standard LLLLL-function of the base change πEÏ€Epi_(E)\pi_{E}Ï€E of πÏ€pi\piÏ€ to GL2(AE)GL2AEGL_(2)(A_(E))\mathrm{GL}_{2}\left(\mathbb{A}_{E}\right)GL2(AE) twisted by χχchi\chiχ,
L(s,πL(s,Ï€L(s,piL(s, \piL(s,Ï€, Ad ))))) is the adjoint LLLLL-function of πÏ€pi\piÏ€,
L(s,μE/F)Ls,μE/FL(s,mu_(E//F))L\left(s, \mu_{E / F}\right)L(s,μE/F) is the Hecke LLLLL-function of the quadratic automorphic character μE/FμE/Fmu_(E//F)\mu_{E / F}μE/F of AF×AF×A_(F)^(xx)\mathbb{A}_{F}^{\times}AF×associated to E/FE/FE//FE / FE/F by class field theory,
αv(fv,χv)αvfv,χvalpha_(v)(f_(v),chi_(v))\alpha_{v}\left(f_{v}, \chi_{v}\right)αv(fv,χv) is a certain normalized local integral of matrix coefficients.
As another example, we consider the 6-dimensional symplectic space W′=W3W′=W3W^(')=W^(3)W^{\prime}=W^{3}W′=W3 over FFFFF. Then the identification W′⊗FV=(W⊗FV)3W′⊗FV=W⊗FV3W^(')ox_(F)V=(Wox_(F)V)^(3)W^{\prime} \otimes_{F} V=\left(W \otimes_{F} V\right)^{3}W′⊗FV=(W⊗FV)3 gives rise to the following seesaw diagram:
Let π1B,π2B,π3BÏ€1B,Ï€2B,Ï€3Bpi_(1)^(B),pi_(2)^(B),pi_(3)^(B)\pi_{1}^{B}, \pi_{2}^{B}, \pi_{3}^{B}Ï€1B,Ï€2B,Ï€3B be irreducible automorphic representations of B×(A)B×(A)B^(xx)(A)B^{\times}(\mathbb{A})B×(A) such that their Jacquet-Langlands transfers π1,π2,π3Ï€1,Ï€2,Ï€3pi_(1),pi_(2),pi_(3)\pi_{1}, \pi_{2}, \pi_{3}Ï€1,Ï€2,Ï€3 to GL2(A)GL2(A)GL_(2)(A)\mathrm{GL}_{2}(\mathbb{A})GL2(A) are cuspidal. Assume that the product of the central characters of π1B,π2B,π3BÏ€1B,Ï€2B,Ï€3Bpi_(1)^(B),pi_(2)^(B),pi_(3)^(B)\pi_{1}^{B}, \pi_{2}^{B}, \pi_{3}^{B}Ï€1B,Ï€2B,Ï€3B is trivial and consider the trilinear period
for decomposable vectors f1∈π1B,f2∈π2B,f3∈π3Bf1∈π1B,f2∈π2B,f3∈π3Bf_(1)inpi_(1)^(B),f_(2)inpi_(2)^(B),f_(3)inpi_(3)^(B)f_{1} \in \pi_{1}^{B}, f_{2} \in \pi_{2}^{B}, f_{3} \in \pi_{3}^{B}f1∈π1B,f2∈π2B,f3∈π3B. Then following the work of HarrisKudla [27] and using the above seesaw diagram, we proved in [32] that
L(s,π1×π2×π3)Ls,Ï€1×π2×π3L(s,pi_(1)xxpi_(2)xxpi_(3))L\left(s, \pi_{1} \times \pi_{2} \times \pi_{3}\right)L(s,Ï€1×π2×π3) is the triple product LLLLL-function of π1,π2,π3Ï€1,Ï€2,Ï€3pi_(1),pi_(2),pi_(3)\pi_{1}, \pi_{2}, \pi_{3}Ï€1,Ï€2,Ï€3,
αv(f1,v,f2,v,f3,v)αvf1,v,f2,v,f3,valpha_(v)(f_(1,v),f_(2,v),f_(3,v))\alpha_{v}\left(f_{1, v}, f_{2, v}, f_{3, v}\right)αv(f1,v,f2,v,f3,v) is a certain normalized local integral of matrix coefficients.
Remark 2.3. The above two formulas are special cases of the Gross-Prasad conjecture [25,26] and its refinement [33]. This conjecture (for special orthogonal groups) was extended to all classical groups by Gan-Gross-Prasad [17], and after the breakthrough of Zhang [63,64], the global conjecture for unitary groups has been proved in a series of works [8-11,62] using the relative trace formula. We should also mention the stunning work of Waldspurger [57-60], which led to the proof of the local Gan-Gross-Prasad conjecture for Bessel models [5-7,45] and Fourier-Jacobi models [2,19] in the ppppp-adic case, where the theta lifting is used to deduce the latter from the former.
3. THE SHIMURA-WALDSPURGER CORRESPONDENCE
In this section, we review some applications of the theta lifting to automorphic forms on metaplectic groups.
3.1. Modular forms of half-integral weight
The theta function
θ(z)=∑n=−∞∞e2πin2zθ(z)=∑n=−∞∞ e2Ï€in2ztheta(z)=sum_(n=-oo)^(oo)e^(2pi in^(2)z)\theta(z)=\sum_{n=-\infty}^{\infty} e^{2 \pi i n^{2} z}θ(z)=∑n=−∞∞e2Ï€in2z
is a modular form of weight 1/21/21//21 / 21/2 and its significance is well known. Thus it is natural to study modular forms of half-integral weight, but Hecke [28, P. 152] realized the difficulty in developing the arithmetic theory; the Hecke operator TnTnT_(n)T_{n}Tn is zero unless nnnnn is a square. In 1973, Shimura [50] revolutionized the theory of modular forms of half-integral weight by relating them to modular forms of integral weight, i.e., he constructed a modular form of weight 2k2k2k2 k2k from a cusp form of weight k+1/2k+1/2k+1//2k+1 / 2k+1/2 by using the converse theorem, where kkkkk is a positive integer. Soon after the discovery of this correspondence, Niwa [46] and Shintani [51] gave an alternative construction using the theta lifting. This was further investigated by Waldspurger [53,56] in the framework of automorphic representations. Namely, he established a correspondence between automorphic representations of Mp2(A)Mp2(A)Mp_(2)(A)\mathrm{Mp}_{2}(\mathbb{A})Mp2(A) (where Mp2(A)Mp2(A)Mp_(2)(A)\mathrm{Mp}_{2}(\mathbb{A})Mp2(A) is the metaplectic cover of SL2(A))SL2(A){:SL_(2)(A))\left.\mathrm{SL}_{2}(\mathbb{A})\right)SL2(A)) and those of PGL2(A)PGL2(A)PGL_(2)(A)\mathrm{PGL}_{2}(\mathbb{A})PGL2(A), which can be viewed as an example of Langlands functoriality.
3.2. Global correspondence
Now we discuss a generalization of the Shimura-Waldspurger correspondence to metaplectic groups of higher rank. Let FFFFF be a number field with adèle ring AAA\mathbb{A}A. We denote by Sp2nSp2nSp_(2n)\mathrm{Sp}_{2 n}Sp2n the symplectic group of rank nnnnn over FFFFF and by Mp2n(A)Mp2nâ¡(A)Mp_(2n)(A)\operatorname{Mp}_{2 n}(\mathbb{A})Mp2nâ¡(A) the metaplectic cover of Sp2n(A)Sp2n(A)Sp_(2n)(A)\mathrm{Sp}_{2 n}(\mathbb{A})Sp2n(A). Recall that this cover splits over Sp2n(F)Sp2n(F)Sp_(2n)(F)\mathrm{Sp}_{2 n}(F)Sp2n(F) canonically, so that we may speak of the unitary representation of Mp2n(A)Mp2nâ¡(A)Mp_(2n)(A)\operatorname{Mp}_{2 n}(\mathbb{A})Mp2nâ¡(A) on the Hilbert space
given by right translation. Since we are interested in genuine automorphic representations of Mp2n(A)Mp2nâ¡(A)Mp_(2n)(A)\operatorname{Mp}_{2 n}(\mathbb{A})Mp2nâ¡(A), i.e., those which do not descend to representations of Sp2n(A)Sp2n(A)Sp_(2n)(A)\mathrm{Sp}_{2 n}(\mathbb{A})Sp2n(A), we only consider its subspace
for the decomposition into the discrete part and the continuous part. Then the theory of Eisenstein series gives an explicit description of Lcont 2(Mp2n)Lcont 2Mp2nL_("cont ")^(2)(Mp_(2n))L_{\text {cont }}^{2}\left(\mathrm{Mp}_{2 n}\right)Lcont 2(Mp2n) in terms of automorphic discrete spectra of proper Levi subgroups of Mp2nMp2nMp_(2n)\mathrm{Mp}_{2 n}Mp2n, i.e., GLn1×⋯×GLnk×Mp2n0GLn1×⋯×GLnk×Mp2n0GL_(n_(1))xx cdots xxGL_(n_(k))xxMp_(2n_(0))\mathrm{GL}_{n_{1}} \times \cdots \times \mathrm{GL}_{n_{k}} \times \mathrm{Mp}_{2 n_{0}}GLn1×⋯×GLnk×Mp2n0 with n1+⋯+nk+n0=nn1+⋯+nk+n0=nn_(1)+cdots+n_(k)+n_(0)=nn_{1}+\cdots+n_{k}+n_{0}=nn1+⋯+nk+n0=n and n0<nn0<nn_(0) < nn_{0}<nn0<n. Thus the problem is to describe the irreducible decomposition of Ldisc 2(Mp2n)Ldisc 2Mp2nL_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n).
To attack this problem, it is better to divide it into two parts as follows:
(1) Describe the decomposition of Ldisc 2(Mp2n)Ldisc 2Mp2nL_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n) into near equivalence classes. Here we say that two irreducible genuine representations π≅⊗vπvπ≅⊗vÏ€vpi~=ox_(v)pi_(v)\pi \cong \otimes_{v} \pi_{v}π≅⊗vÏ€v and π′≅⊗vπv′π′≅⊗vÏ€v′pi^(')~=ox_(v)pi_(v)^(')\pi^{\prime} \cong \otimes_{v} \pi_{v}^{\prime}π′≅⊗vÏ€v′ of Mp2n(A)Mp2nâ¡(A)Mp_(2n)(A)\operatorname{Mp}_{2 n}(\mathbb{A})Mp2nâ¡(A) are nearly equivalent if πvÏ€vpi_(v)\pi_{v}Ï€v and πv′Ï€v′pi_(v)^(')\pi_{v}^{\prime}Ï€v′ are equivalent for almost all places vvvvv of FFFFF. (In particular, if πÏ€pi\piÏ€ and π′π′pi^(')\pi^{\prime}π′ are equivalent, then they are nearly equivalent.) Note that πvÏ€vpi_(v)\pi_{v}Ï€v is unramified for almost all vvvvv, so that it determines and is determined by a semisimple conjugacy class cψv(πv)cψvÏ€vc_(psi_(v))(pi_(v))c_{\psi_{v}}\left(\pi_{v}\right)cψv(Ï€v) in Sp2n(C)Sp2n(C)Sp_(2n)(C)\mathrm{Sp}_{2 n}(\mathbb{C})Sp2n(C) (depending on a choice of a nontrivial additive character ψvψvpsi_(v)\psi_{v}ψv of FvFvF_(v)F_{v}Fv ) via the Satake isomorphism. In other words, the near equivalence classes of irreducible genuine representations of Mp2n(A)Mp2n(A)Mp_(2n)(A)\mathrm{Mp}_{2 n}(\mathbb{A})Mp2n(A) can be parametrized by families of semisimple conjugacy classes
{cv}vcvv{c_(v)}_(v)\left\{c_{v}\right\}_{v}{cv}v
in Sp2n(C)Sp2n(C)Sp_(2n)(C)\mathrm{Sp}_{2 n}(\mathbb{C})Sp2n(C), where we identify two families if they are equal for almost all vvvvv. Thus we want to describe the families {cv}vcvv{c_(v)}_(v)\left\{c_{v}\right\}_{v}{cv}v which correspond to the near equivalence classes in Ldisc 2(Mp2n)Ldisc 2Mp2nL_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n).
(2) Describe the irreducible decomposition of each near equivalence class. Namely, for any near equivalence class CCCCC in Ldisc 2(Mp2n)Ldisc 2Mp2nL_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n) and any irreducible genuine representation πÏ€pi\piÏ€ of Mp2n(A)Mp2nâ¡(A)Mp_(2n)(A)\operatorname{Mp}_{2 n}(\mathbb{A})Mp2nâ¡(A), we want to give an explicit formula for the multiplicity of πÏ€pi\piÏ€ in CCCCC in terms of the classification of representations.
In [20], we solved (1) completely and (2) partially; we described the families {cv}vcvv{c_(v)}_(v)\left\{c_{v}\right\}_{v}{cv}v as above in terms of automorphic representations of general linear groups, and admitting that Arthur's endoscopic classification [1] can be extended to nonsplit odd special orthogonal groups, we established the multiplicity formula for the tempered part of Ldisc 2(Mp2n)Ldisc 2Mp2nL_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n).
Now we state the first result precisely.
Theorem 3.1 ([20]). Fix a nontrivial additive character ψ=⊗vψvψ=⊗vψvpsi=ox_(v)psi_(v)\psi=\otimes_{v} \psi_{v}ψ=⊗vψv of A/FA/FA//F\mathbb{A} / FA/F. Then we have a decomposition
where ϕÏ•phi\phiÏ• runs over elliptic A-parameters for Mp2nMp2nMp_(2n)\mathrm{Mp}_{2 n}Mp2n. Here an elliptic A-parameter for Mp2nMp2nMp_(2n)\mathrm{Mp}_{2 n}Mp2n is defined to be a formal finite direct sum
(which is a substitute for a hypothetical L-homomorphism LF×SL2(C)→Sp2n(C)LF×SL2(C)→Sp2n(C)L_(F)xxSL_(2)(C)rarrSp_(2n)(C)\mathscr{L}_{F} \times \mathrm{SL}_{2}(\mathbb{C}) \rightarrow \mathrm{Sp}_{2 n}(\mathbb{C})LF×SL2(C)→Sp2n(C) ), where
ϕiÏ•iphi_(i)\phi_{i}Ï•i is an irreducible self-dual cuspidal automorphic representation of GLni(A)GLni(A)GL_(n_(i))(A)\mathrm{GL}_{n_{i}}(\mathbb{A})GLni(A) (which is hypothetically identified with an ninin_(i)n_{i}ni-dimensional irreducible representation of LFLFL_(F)\mathscr{L}_{F}LF ),
SdiSdiS_(d_(i))S_{d_{i}}Sdi is the didid_(i)d_{i}di-dimensional irreducible representation of SL2(C)SL2(C)SL_(2)(C)\mathrm{SL}_{2}(\mathbb{C})SL2(C),
if didid_(i)d_{i}di is odd, then ϕiÏ•iphi_(i)\phi_{i}Ï•i is symplectic, i.e., the exterior square LLLLL-function L(s,ϕi,∧2)Ls,Ï•i,∧2L(s,phi_(i),^^^(2))L\left(s, \phi_{i}, \wedge^{2}\right)L(s,Ï•i,∧2) has a pole at s=1s=1s=1s=1s=1 (and hence ninin_(i)n_{i}ni is even),
if didid_(i)d_{i}di is even, then ϕiÏ•iphi_(i)\phi_{i}Ï•i is orthogonal, i.e., the symmetric square L-function L(s,ϕiLs,Ï•iL(s,phi_(i):}L\left(s, \phi_{i}\right.L(s,Ï•i, Sym 2)2{:^(2))\left.^{2}\right)2) has a pole at s=1s=1s=1s=1s=1,
if i≠ji≠ji!=ji \neq ji≠j, then (ϕi,di)≠(ϕj,dj)Ï•i,di≠ϕj,dj(phi_(i),d_(i))!=(phi_(j),d_(j))\left(\phi_{i}, d_{i}\right) \neq\left(\phi_{j}, d_{j}\right)(Ï•i,di)≠(Ï•j,dj),
Also, Lϕ2(Mp2n)LÏ•2Mp2nL_(phi)^(2)(Mp_(2n))L_{\phi}^{2}\left(\mathrm{Mp}_{2 n}\right)LÏ•2(Mp2n) is defined as the near equivalence class in Ldisc 2(Mp2n)Ldisc 2Mp2nL_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n) which corresponds to the family of semisimple conjugacy classes
in Sp2n(C)Sp2n(C)Sp_(2n)(C)\mathrm{Sp}_{2 n}(\mathbb{C})Sp2n(C) given as follows (so that any irreducible summand πÏ€pi\piÏ€ of Lϕ2(Mp2n)LÏ•2Mp2nL_(phi)^(2)(Mp_(2n))L_{\phi}^{2}\left(\mathrm{Mp}_{2 n}\right)LÏ•2(Mp2n) satisfies cψv(πv)=cv(ϕv)cψvÏ€v=cvÏ•vc_(psi_(v))(pi_(v))=c_(v)(phi_(v))c_{\psi_{v}}\left(\pi_{v}\right)=c_{v}\left(\phi_{v}\right)cψv(Ï€v)=cv(Ï•v) for almost all vvvvv ). Suppose that vvvvv is finite and ϕi,vÏ•i,vphi_(i,v)\phi_{i, v}Ï•i,v is unramified for all iiiii. Let cv(ϕi,v)cvÏ•i,vc_(v)(phi_(i,v))c_{v}\left(\phi_{i, v}\right)cv(Ï•i,v) be the semisimple conjugacy class in GLni(C)GLni(C)GL_(n_(i))(C)\mathrm{GL}_{n_{i}}(\mathbb{C})GLni(C) which corresponds to ϕi,vÏ•i,vphi_(i,v)\phi_{i, v}Ï•i,v and put
for any positive integer ddddd, where qvqvq_(v)q_{v}qv is the cardinality of the residue field of FvFvF_(v)F_{v}Fv. We regard cv(ϕi,v)⊗Qv(di)cvÏ•i,v⊗Qvdic_(v)(phi_(i,v))oxQ_(v)(d_(i))c_{v}\left(\phi_{i, v}\right) \otimes Q_{v}\left(d_{i}\right)cv(Ï•i,v)⊗Qv(di) as a semisimple conjugacy class in Spnidi(C)Spnidi(C)Sp_(n_(i)d_(i))(C)\mathrm{Sp}_{n_{i} d_{i}}(\mathbb{C})Spnidi(C). Then we set
To state the second result precisely, we need to introduce more notation. For each place vvvvv of FFFFF, let WFvWFvW_(F_(v))\mathcal{W}_{F_{v}}WFv be the Weil group of FvFvF_(v)F_{v}Fv and put
LFv={WFv if v is infinite WFv×SL2(C) if v is finite LFv=WFv if v is infinite WFv×SL2(C) if v is finite L_(F_(v))={[W_(F_(v))," if "v" is infinite "],[W_(F_(v))xxSL_(2)(C)," if "v" is finite "]:}\mathscr{L}_{F_{v}}= \begin{cases}\mathcal{W}_{F_{v}} & \text { if } v \text { is infinite } \\ \mathcal{W}_{F_{v}} \times \mathrm{SL}_{2}(\mathbb{C}) & \text { if } v \text { is finite }\end{cases}LFv={WFv if v is infinite WFv×SL2(C) if v is finiteÂ
Let ϕ=⨁iϕi⊗SdiÏ•=â¨i ϕi⊗Sdiphi=bigoplus_(i)phi_(i)oxS_(d_(i))\phi=\bigoplus_{i} \phi_{i} \otimes S_{d_{i}}Ï•=â¨iÏ•i⊗Sdi be an elliptic AAAAA-parameter for Mp2nMp2nMp_(2n)\mathrm{Mp}_{2 n}Mp2n. We regard its local component ϕv=⨁iϕi,v⊗SdiÏ•v=â¨i ϕi,v⊗Sdiphi_(v)=bigoplus_(i)phi_(i,v)oxS_(d_(i))\phi_{v}=\bigoplus_{i} \phi_{i, v} \otimes S_{d_{i}}Ï•v=â¨iÏ•i,v⊗Sdi at vvvvv as a local AAAAA-parameter
for almost all vvvvv, where FrvFrvFr_(v)\operatorname{Fr}_{v}Frv is a Frobenius element at vvvvv. We denote by ℑϕvâ„‘Ï•vâ„‘_(phi_(v))\Im_{\phi_{v}}â„‘Ï•v the component group of the centralizer of ϕvÏ•vphi_(v)\phi_{v}Ï•v in Sp2n(C)Sp2n(C)Sp_(2n)(C)\mathrm{Sp}_{2 n}(\mathbb{C})Sp2n(C), which is an elementary abelian 2-group, and by
the global component group of ϕÏ•phi\phiÏ•, which is formally defined as an elementary abelian 2-group with a basis {ai}iaii{a_(i)}_(i)\left\{a_{i}\right\}_{i}{ai}i indexed by {ϕi⊗Sdi}iÏ•i⊗Sdii{phi_(i)oxS_(d_(i))}_(i)\left\{\phi_{i} \otimes S_{d_{i}}\right\}_{i}{Ï•i⊗Sdi}i. Then we have a natural homomorphism Sϕ→SϕvSϕ→SÏ•vS_(phi)rarrS_(phi_(v))S_{\phi} \rightarrow S_{\phi_{v}}Sϕ→SÏ•v for all vvvvv. We also consider the compact abelian group Sϕ,A=∏vςϕvSÏ•,A=âˆv ςϕvS_(phi,A)=prod_(v)Ï‚_(phi_(v))S_{\phi, \mathbb{A}}=\prod_{v} \varsigma_{\phi_{v}}SÏ•,A=âˆvςϕv and the diagonal map
where η=⊗vηvη=⊗vηveta=ox_(v)eta_(v)\eta=\otimes_{v} \eta_{v}η=⊗vηv runs over continuous characters of Sϕ,ASÏ•,AS_(phi,A)S_{\phi, \mathbb{A}}SÏ•,A. Here πηπηpi_(eta)\pi_{\eta}πη is defined as the restricted tensor product of representations πηvπηvpi_(eta_(v))\pi_{\eta_{v}}πηv in the local L-packets
associated to ϕvÏ•vphi_(v)\phi_{v}Ï•v (depending on ψvψvpsi_(v)\psi_{v}ψv ), which consist of irreducible genuine representations of Mp2n(Fv)Mp2nâ¡FvMp_(2n)(F_(v))\operatorname{Mp}_{2 n}\left(F_{v}\right)Mp2nâ¡(Fv) indexed by characters of SϕvSÏ•vS_(phi_(v))S_{\phi_{v}}SÏ•v. Also, if we define a character εϕεϕepsi_(phi)\varepsilon_{\phi}εϕ of ℑϕℑϕℑ_(phi)\Im_{\phi}â„‘Ï• by
where ε(s,ϕi)εs,Ï•iepsi(s,phi_(i))\varepsilon\left(s, \phi_{i}\right)ε(s,Ï•i) is the standard εεepsi\varepsilonε-function of ϕiÏ•iphi_(i)\phi_{i}Ï•i, then mηmηm_(eta)m_{\eta}mη is given by
Remark 3.3. In fact, we gave another proof of the result of Waldspurger for Mp2Mp2Mp_(2)\mathrm{Mp}_{2}Mp2 [53, 56], noting that an irreducible cuspidal automorphic representation of GL2(A)GL2(A)GL_(2)(A)\mathrm{GL}_{2}(\mathbb{A})GL2(A) is symplectic if and only if its central character is trivial.
Remark 3.4. If we denote by SO2n+1SO2n+1SO_(2n+1)\mathrm{SO}_{2 n+1}SO2n+1 the split odd special orthogonal group of rank nnnnn over FFFFF and by Ldisc 2(SO2n+1)Ldisc 2SO2n+1L_("disc ")^(2)(SO_(2n+1))L_{\text {disc }}^{2}\left(\mathrm{SO}_{2 n+1}\right)Ldisc 2(SO2n+1) the discrete part of L2(SO2n+1(F)∖SO2n+1(A))L2SO2n+1(F)∖SO2n+1(A)L^(2)(SO_(2n+1)(F)\\SO_(2n+1)(A))L^{2}\left(\mathrm{SO}_{2 n+1}(F) \backslash \mathrm{SO}_{2 n+1}(\mathbb{A})\right)L2(SO2n+1(F)∖SO2n+1(A)), then the decomposition of Ldisc 2(Mp2n)Ldisc 2Mp2nL_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n) is similar to that of Ldisc 2(SO2n+1)Ldisc 2SO2n+1L_("disc ")^(2)(SO_(2n+1))L_{\text {disc }}^{2}\left(\mathrm{SO}_{2 n+1}\right)Ldisc 2(SO2n+1) given by Arthur [1], except that the condition η∘Δ=εϕη∘Δ=εϕeta@Delta=epsi_(phi)\eta \circ \Delta=\varepsilon_{\phi}η∘Δ=εϕ in the former has to be replaced by η∘Δ=1η∘Δ=1eta@Delta=1\eta \circ \Delta=1η∘Δ=1 in the latter.
In the proof of his result for n=1n=1n=1n=1n=1, Waldspurger used the theta lifting between Mp2Mp2Mp_(2)\mathrm{Mp}_{2}Mp2 and (inner forms of) PGL2≅SO3PGL2≅SO3PGL_(2)~=SO_(3)\mathrm{PGL}_{2} \cong \mathrm{SO}_{3}PGL2≅SO3. Thus, in general, it would be natural to use the theta lifting between Mp2nMp2nMp_(2n)\mathrm{Mp}_{2 n}Mp2n and (inner forms of) SO2n+1SO2n+1SO_(2n+1)\mathrm{SO}_{2 n+1}SO2n+1, and then transfer Arthur's endoscopic classification from SO2n+1SO2n+1SO_(2n+1)\mathrm{SO}_{2 n+1}SO2n+1 to Mp2nMp2nMp_(2n)\mathrm{Mp}_{2 n}Mp2n. However, there is a serious obstacle in this approach. Indeed, if πÏ€pi\piÏ€ is an irreducible genuine cuspidal automorphic representation of Mp2n(A)Mp2n(A)Mp_(2n)(A)\mathrm{Mp}_{2 n}(\mathbb{A})Mp2n(A) and its standard LLLLL-function L(s,π)L(s,Ï€)L(s,pi)L(s, \pi)L(s,Ï€) vanishes at s=1/2s=1/2s=1//2s=1 / 2s=1/2, then the theta lift of πÏ€pi\piÏ€ to SO2n+1(A)SO2n+1(A)SO_(2n+1)(A)\mathrm{SO}_{2 n+1}(\mathbb{A})SO2n+1(A) is zero. When n=1n=1n=1n=1n=1, Waldspurger proved that the twisted standard LLLLL-function L(s,π,χ)L(s,Ï€,χ)L(s,pi,chi)L(s, \pi, \chi)L(s,Ï€,χ) does not vanish at s=1/2s=1/2s=1//2s=1 / 2s=1/2 for some quadratic automorphic character χχchi\chiχ of A×A×A^(xx)\mathbb{A}^{\times}A×and could use the twisted theta lifting to establish the desired correspondence. But for general nnnnn, the existence of such a character χχchi\chiχ is considered extremely difficult to prove.
To circumvent this difficulty, we used the theta lifting in the so-called stable range studied by LiLiLi\mathrm{Li}Li [44]. More precisely, for any irreducible genuine representation πÏ€pi\piÏ€ of Mp2n(A)Mp2n(A)Mp_(2n)(A)\mathrm{Mp}_{2 n}(\mathbb{A})Mp2n(A), we consider its (abstract) theta lift
to SO2r+1(A)SO2r+1(A)SO_(2r+1)(A)\mathrm{SO}_{2 r+1}(\mathbb{A})SO2r+1(A) with r≫2nr≫2nr≫2nr \gg 2 nr≫2n. Then it follows from the result of LiLiLi\mathrm{Li}Li that if πÏ€pi\piÏ€ occurs in Ldisc 2(Mp2n)Ldisc 2Mp2nL_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n), then θabs(π)θabs(Ï€)theta^(abs)(pi)\theta^{\mathrm{abs}}(\pi)θabs(Ï€) occurs in Ldisc 2(SO2r+1)Ldisc 2SO2r+1L_("disc ")^(2)(SO_(2r+1))L_{\text {disc }}^{2}\left(\mathrm{SO}_{2 r+1}\right)Ldisc 2(SO2r+1). Combining this with the analytic theory of standard LLLLL-functions, we may deduce Theorem 3.1 from Arthur's endoscopic classification for SO2r+1SO2r+1SO_(2r+1)\mathrm{SO}_{2 r+1}SO2r+1. Moreover, if πÏ€pi\piÏ€ is an irreducible summand of the tempered part of Ldisc 2(Mp2n)Ldisc 2Mp2nL_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n), then we proved that
where m(⋅)m(â‹…)m(*)m(\cdot)m(â‹…) denotes the multiplicity in the automorphic discrete spectrum. (We expect that this equality holds for any irreducible summand πÏ€pi\piÏ€ of Ldisc 2(Mp2n)Ldisc 2Mp2nL_("disc ")^(2)(Mp_(2n))L_{\text {disc }}^{2}\left(\mathrm{Mp}_{2 n}\right)Ldisc 2(Mp2n).) Using this and describing the local theta lifting between Mp2nMp2nMp_(2n)\mathrm{Mp}_{2 n}Mp2n and SO2r+1SO2r+1SO_(2r+1)\mathrm{SO}_{2 r+1}SO2r+1 explicitly, we may deduce Theorem 3.2.
Remark 3.5. When n=2n=2n=2n=2n=2 and ϕÏ•phi\phiÏ• is nontempered, we proved a similar decomposition of Lϕ2(Mp4)LÏ•2Mp4L_(phi)^(2)(Mp_(4))L_{\phi}^{2}\left(\mathrm{Mp}_{4}\right)LÏ•2(Mp4) in [21]. Note that πηπηpi_(eta)\pi_{\eta}πη is not necessarily irreducible and εϕεϕepsi_(phi)\varepsilon_{\phi}εϕ has to be modified in this case.
3.3. Local correspondence
There is a local analog of the above correspondence, called the local Shimura correspondence. For simplicity, we only consider the ppppp-adic case and write FFFFF for a finite extension of QpQpQ_(p)\mathbb{Q}_{p}Qp. We are interested in the set
of equivalence classes of irreducible genuine representations of the metaplectic group Mp2n(F)Mp2nâ¡(F)Mp_(2n)(F)\operatorname{Mp}_{2 n}(F)Mp2nâ¡(F). Recall that there are precisely two (2n+1)(2n+1)(2n+1)(2 n+1)(2n+1)-dimensional quadratic spaces V+V+V^(+)V^{+}V+ and V−V−V^(-)V^{-}V−over FFFFF with trivial discriminant (up to isometry). Let SO(V+)SOâ¡V+SO(V^(+))\operatorname{SO}\left(V^{+}\right)SOâ¡(V+)and SO(V−)SOâ¡V−SO(V^(-))\operatorname{SO}\left(V^{-}\right)SOâ¡(V−)denote the special orthogonal groups of V+V+V^(+)V^{+}V+and V−V−V^(-)V^{-}V−, respectively. Then the local Shimura correspondence, which was established by Gan-Savin [22] in the ppppp-adic case, says that there is a bijection (depending on a choice of a nontrivial additive character ψψpsi\psiψ of FFFFF )
given by the local theta lifting. Namely, for any irreducible genuine representation πÏ€pi\piÏ€ of Mp2n(F),θ(π)Mp2nâ¡(F),θ(Ï€)Mp_(2n)(F),theta(pi)\operatorname{Mp}_{2 n}(F), \theta(\pi)Mp2nâ¡(F),θ(Ï€) is defined as the unique irreducible representation of SO(Vε)SOâ¡VεSO(V^(epsi))\operatorname{SO}\left(V^{\varepsilon}\right)SOâ¡(Vε) with the unique sign ε=±Îµ=±epsi=+-\varepsilon= \pmε=± such that
where ωεωεomega^(epsi)\omega^{\varepsilon}ωε is the Weil representation of Mp2n(F)×SO(Vε)Mp2nâ¡(F)×SOâ¡VεMp_(2n)(F)xx SO(V^(epsi))\operatorname{Mp}_{2 n}(F) \times \operatorname{SO}\left(V^{\varepsilon}\right)Mp2nâ¡(F)×SOâ¡(Vε) (depending on ψψpsi\psiψ ). Moreover, they proved various natural properties:
θθtheta\thetaθ preserves the square-integrability,
θθtheta\thetaθ preserves the temperedness,
θθtheta\thetaθ is compatible with the theory of RRRRR-groups,
θθtheta\thetaθ is compatible with the Langlands classification,
and used θθtheta\thetaθ to transfer the local Langlands correspondence from SO(Vε)SOâ¡VεSO(V^(epsi))\operatorname{SO}\left(V^{\varepsilon}\right)SOâ¡(Vε) to Mp2n(F)Mp2nâ¡(F)Mp_(2n)(F)\operatorname{Mp}_{2 n}(F)Mp2nâ¡(F). (In particular, this defines the local LLLLL-packets in the statement of Theorem 3.2.)
Remark 3.6. The local theta lifting has also been described for other reductive dual pairs in terms of the local Langlands correspondence. See [3,4,19][3,4,19][3,4,19][3,4,19][3,4,19] for recent progress.
As in Section 2.3, the local theta lifting can produce various relations between local analogs of periods. For example, we consider an irreducible genuine square-integrable representation πÏ€pi\piÏ€ of Mp2n(F)Mp2nâ¡(F)Mp_(2n)(F)\operatorname{Mp}_{2 n}(F)Mp2nâ¡(F) and its formal degree d(π)d(Ï€)d(pi)d(\pi)d(Ï€). Recall that d(π)d(Ï€)d(pi)d(\pi)d(Ï€) is defined as the positive real number for which the Schur orthogonality relation
which is a local analog of the Rallis inner product formula [47], by using the doubling seesaw diagram:
Remark 3.7. Recall from Section 2.3 that some automorphic periods can be expressed in terms of special values of LLLLL-functions. Similarly, formal degrees should be expressed in terms of arithmetic invariants as follows. Let GGGGG be a connected reductive group over FFFFF. For simplicity, we assume that GGGGG is a pure inner form of a quasisplit group and the center of GGGGG is anisotropic. Let πÏ€pi\piÏ€ be an irreducible square-integrable representation of G(F)G(F)G(F)G(F)G(F). Let d(π)d(Ï€)d(pi)d(\pi)d(Ï€) denote the formal degree of πÏ€pi\piÏ€ with respect to the Haar measure on G(F)G(F)G(F)G(F)G(F) determined by a Chevalley basis of the Lie algebra of the split form of GGGGG and a fixed nontrivial additive character ψψpsi\psiψ of FFFFF. Then the formal degree conjecture [29] says that
ϕ:LF→LGÏ•:LF→LGphi:L_(F)rarr^(L)G\phi: \mathscr{L}_{F} \rightarrow{ }^{L} GÏ•:LF→LG is the LLLLL-parameter (conjecturally) associated to πÏ€pi\piÏ€,
ςϕςϕς_(phi)\varsigma_{\phi}ςϕ is the component group of the centralizer of ϕÏ•phi\phiÏ• in G^G^hat(G)\hat{G}G^,
ηηeta\etaη is the irreducible representation of ςϕςϕς_(phi)\varsigma_{\phi}ςϕ (conjecturally) associated to πÏ€pi\piÏ€,
Ad is the adjoint representation of LGLG^(L)G{ }^{L} GLG on its Lie algebra,
γ(sγ(sgamma(s\gamma(sγ(s, Ad ∘ϕ,ψ)∘ϕ,ψ)@phi,psi)\circ \phi, \psi)∘ϕ,ψ) is the local γγgamma\gammaγ-factor given by
In [34], we proved this conjecture for (inner forms of) SO2n+1SO2n+1SO_(2n+1)\mathrm{SO}_{2 n+1}SO2n+1 and its analog for Mp2nMp2nMp_(2n)\mathrm{Mp}_{2 n}Mp2n by using the main identity of Lapid-Mao [43] and the above relation between formal degrees.
4. GEOMETRIC REALIZATION OF THE JACOUET-LANGLANDS CORRESPONDENCE
Let πÏ€pi\piÏ€ be an irreducible automorphic representation of G(A)G(A)G(A)G(\mathbb{A})G(A) and suppose that πÏ€pi\piÏ€ is cohomological, so that πÏ€pi\piÏ€ occurs in the cohomology H∗(X,C)H∗(X,C)H^(**)(X,C)H^{*}(X, \mathbb{C})H∗(X,C), where GGGGG is a connected reductive group and XXXXX is a locally symmetric space for GGGGG. Then it is natural to ask whether functorial transfers of πÏ€pi\piÏ€ can be realized geometrically. In this section, we discuss the simplest example, i.e., the Jacquet-Langlands correspondence for GL2GL2GL_(2)\mathrm{GL}_{2}GL2 and its inner forms.
Let FFFFF be a totally real number field. Let Aand AfAand AfA^("and ")A_(f)\mathbb{A}^{\text {and }} \mathbb{A}_{f}Aand Af denote the rings of adèles and finite adèles of FFFFF, respectively. Let πÏ€pi\piÏ€ be an irreducible cuspidal automorphic representation of GL2(A)GL2(A)GL_(2)(A)\mathrm{GL}_{2}(\mathbb{A})GL2(A) such that πvÏ€vpi_(v)\pi_{v}Ï€v is the discrete series of weight 2 for all infinite places vvvvv of FFFFF. For simplicity, we assume that the central character of πÏ€pi\piÏ€ is trivial, the level of πÏ€pi\piÏ€ is square-free,
and the Hecke eigenvalues of πÏ€pi\piÏ€ lie in QQQ\mathbb{Q}Q. Let BBBBB be a quaternion division algebra over FFFFF. For each place vvvvv of FFFFF, put Bv=B⊗FFvBv=B⊗FFvB_(v)=Box_(F)F_(v)B_{v}=B \otimes_{F} F_{v}Bv=B⊗FFv. Let VBVBV_(B)\mathcal{V}_{B}VB be the set of infinite places vvvvv of FFFFF such that BvBvB_(v)B_{v}Bv is split. Assume that VB≠∅VB≠∅V_(B)!=O/\mathcal{V}_{B} \neq \varnothingVB≠∅ and put d=|VB|d=VBd=|V_(B)|d=\left|\mathcal{V}_{B}\right|d=|VB|. We denote by XBXBX_(B)X_{B}XB the Shimura variety for B×B×B^(xx)B^{\times}B×(with respect to some neat open compact subgroup KfKfK_(f)K_{f}Kf of B×(Af)B×AfB^(xx)(A_(f))B^{\times}\left(\mathbb{A}_{f}\right)B×(Af) ), which is a ddddd-dimensional smooth projective variety over the reflex field F′F′F^(')F^{\prime}F′, so that
F′F′F^(')F^{\prime}F′ is the number field contained in CCC\mathbb{C}C such that
where S±S±S^(+-)\mathfrak{S}^{ \pm}S±is the union of the upper and lower half-planes.
Now assume that the Jacquet-Langlands transfer πBÏ€Bpi^(B)\pi^{B}Ï€B of πÏ€pi\piÏ€ to B×(A)B×(A)B^(xx)(A)B^{\times}(\mathbb{A})B×(A) exists, which is the case if and only if πvÏ€vpi_(v)\pi_{v}Ï€v is a discrete series for all vvvvv at which BBBBB is ramified, and that KfKfK_(f)K_{f}Kf is chosen appropriately so that dim(πfB)Kf=1dimâ¡Ï€fBKf=1dim (pi_(f)^(B))^(K_(f))=1\operatorname{dim}\left(\pi_{f}^{B}\right)^{K_{f}}=1dimâ¡(Ï€fB)Kf=1. Here πfBÏ€fBpi_(f)^(B)\pi_{f}^{B}Ï€fB is the finite component of πBÏ€Bpi^(B)\pi^{B}Ï€B and (πfB)KfÏ€fBKf(pi_(f)^(B))^(K_(f))\left(\pi_{f}^{B}\right)^{K_{f}}(Ï€fB)Kf is the space of KfKfK_(f)K_{f}Kf-fixed vectors in πfBÏ€fBpi_(f)^(B)\pi_{f}^{B}Ï€fB. Then it follows from Matsushima's formula that πBÏ€Bpi_(B)\pi_{B}Ï€B occurs in the cohomology H∗(XB,C)H∗XB,CH^(**)(X_(B),C)H^{*}\left(X_{B}, \mathbb{C}\right)H∗(XB,C). More precisely, we consider the rational cohomology H∗(XB,Q)H∗XB,QH^(**)(X_(B),Q)H^{*}\left(X_{B}, \mathbb{Q}\right)H∗(XB,Q) and its πÏ€pi\piÏ€-isotypic component
H∗(XB,Q)π={α∈H∗(XB,Q)∣Tvα=χπv(Tv)α for all Tv∈Hv and almost all v}H∗XB,QÏ€=α∈H∗XB,Q∣Tvα=χπvTvα for all Tv∈Hv and almost all vH^(**)(X_(B),Q)_(pi)={alpha inH^(**)(X_(B),Q)∣T_(v)alpha=chi_(pi_(v))(T_(v))alpha" for all "T_(v)inH_(v)" and almost all "v}H^{*}\left(X_{B}, \mathbb{Q}\right)_{\pi}=\left\{\alpha \in H^{*}\left(X_{B}, \mathbb{Q}\right) \mid T_{v} \alpha=\chi_{\pi_{v}}\left(T_{v}\right) \alpha \text { for all } T_{v} \in \mathscr{H}_{v} \text { and almost all } v\right\}H∗(XB,Q)Ï€={α∈H∗(XB,Q)∣Tvα=χπv(Tv)α for all Tv∈Hv and almost all v}
where Hv=Q[Kv∖Bv×/Kv]Hv=QKv∖Bv×/KvH_(v)=Q[K_(v)\\B_(v)^(xx)//K_(v)]\mathscr{H}_{v}=\mathbb{Q}\left[K_{v} \backslash B_{v}^{\times} / K_{v}\right]Hv=Q[Kv∖Bv×/Kv] is the Hecke algebra with respect to the standard maximal compact subgroup KvKvK_(v)K_{v}Kv of Bv×≅GL2(Fv)Bv×≅GL2FvB_(v)^(xx)~=GL_(2)(F_(v))B_{v}^{\times} \cong \mathrm{GL}_{2}\left(F_{v}\right)Bv×≅GL2(Fv) and χπv:Hv→Qχπv:Hv→Qchi_(pi_(v)):H_(v)rarrQ\chi_{\pi_{v}}: \mathscr{H}_{v} \rightarrow \mathbb{Q}χπv:Hv→Q is the character by which HvHvH_(v)\mathscr{H}_{v}Hv acts on πvKvÏ€vKvpi_(v)^(K_(v))\pi_{v}^{K_{v}}Ï€vKv. Then H∗(XB,Q)πH∗XB,QÏ€H^(**)(X_(B),Q)_(pi)H^{*}\left(X_{B}, \mathbb{Q}\right)_{\pi}H∗(XB,Q)Ï€ is concentrated in the middle degree ddddd and Hd(XB,Q)πHdXB,QÏ€H^(d)(X_(B),Q)_(pi)H^{d}\left(X_{B}, \mathbb{Q}\right)_{\pi}Hd(XB,Q)Ï€ is a 2d2d2^(d)2^{d}2d-dimensional vector space over QQQ\mathbb{Q}Q. Moreover, for any prime ℓâ„“â„“\ellâ„“, the ℓâ„“â„“\ellâ„“-adic representation of Gal(Q¯/F′)Galâ¡Q¯/F′Gal( bar(Q)//F^('))\operatorname{Gal}\left(\overline{\mathbb{Q}} / F^{\prime}\right)Galâ¡(Q¯/F′) on
is given by the so-called tensor induction of ρπ,ℓÏÏ€,â„“rho_(pi,â„“)\rho_{\pi, \ell}ÏÏ€,â„“, where ρπ,ℓÏÏ€,â„“rho_(pi,â„“)\rho_{\pi, \ell}ÏÏ€,â„“ is the 2 -dimensional ℓâ„“â„“\ellâ„“-adic representation associated to πÏ€pi\piÏ€. We remark that it depends only on ρπ,ℓÏÏ€,â„“rho_(pi,â„“)\rho_{\pi, \ell}ÏÏ€,â„“ and VBVBV_(B)\mathcal{V}_{B}VB.
Suppose that we have two quaternion algebras B1B1B_(1)B_{1}B1 and B2B2B_(2)B_{2}B2 as above such that
Write X1=XB1X1=XB1X_(1)=X_(B_(1))X_{1}=X_{B_{1}}X1=XB1 and X2=XB2X2=XB2X_(2)=X_(B_(2))X_{2}=X_{B_{2}}X2=XB2 for the corresponding Shimura varieties, which are of the same dimension d=|VB1|=|VB2|d=VB1=VB2d=|V_(B_(1))|=|V_(B_(2))|d=\left|\mathcal{V}_{B_{1}}\right|=\left|\mathcal{V}_{B_{2}}\right|d=|VB1|=|VB2| over the same reflex field F′F′F^(')F^{\prime}F′. Then we have
of ℓâ„“â„“\ellâ„“-adic representations of Gal(Q¯/F′)Galâ¡Q¯/F′Gal( bar(Q)//F^('))\operatorname{Gal}\left(\overline{\mathbb{Q}} / F^{\prime}\right)Galâ¡(Q¯/F′) for all ℓâ„“â„“\ellâ„“.
Conjecturally, these isomorphisms are obtained from a single isomorphism
given as follows. By (2) and the Künneth formula, the space of Gal(Q¯/F′)Galâ¡Q¯/F′Gal( bar(Q)//F^('))\operatorname{Gal}\left(\overline{\mathbb{Q}} / F^{\prime}\right)Galâ¡(Q¯/F′)-fixed vectors in
is nonzero. Hence the Tate conjecture predicts the existence of an algebraic cycle Z∈CHd(X1×X2)Z∈CHdX1×X2Z inCH^(d)(X_(1)xxX_(2))Z \in \mathrm{CH}^{d}\left(X_{1} \times X_{2}\right)Z∈CHd(X1×X2) which realizes (1) and (2). Namely, let p1p1p_(1)p_{1}p1 and p2p2p_(2)p_{2}p2 be the two projections
ιZ⊗idCιZ⊗idCiota_(Z)oxid_(C)\iota_{Z} \otimes \operatorname{id}_{\mathbb{C}}ιZ⊗idC preserves the Hodge decomposition,
ιZ⊗idQℓιZ⊗idQâ„“iota_(Z)oxid_(Q_(â„“))\iota_{Z} \otimes \operatorname{id}_{\mathbb{Q}_{\ell}}ιZ⊗idQâ„“ is Gal(Q¯/F′)Galâ¡Q¯/F′Gal( bar(Q)//F^('))\operatorname{Gal}\left(\overline{\mathbb{Q}} / F^{\prime}\right)Galâ¡(Q¯/F′)-equivariant for all ℓâ„“â„“\ellâ„“.
When d=1d=1d=1d=1d=1, the existence of ZZZZZ in fact follows from the result of Faltings [16]. But for general ddddd, this remains an open problem. On the other hand, noting that the Hodge conjecture reduces it to finding a Hodge cycle on X1×X2X1×X2X_(1)xxX_(2)X_{1} \times X_{2}X1×X2, i.e., an element in
Theorem 4.1 ([35]). Assume that B1B1B_(1)B_{1}B1 and B2B2B_(2)B_{2}B2 are ramified at some infinite place vvvvv of FFFFF. Then there exists a Hodge cycle ξξxi\xiξ on X1×X2X1×X2X_(1)xxX_(2)X_{1} \times X_{2}X1×X2 which induces an isomorphism
ιξ⊗idCιξ⊗idCiota xi oxid_(C)\iota \xi \otimes \mathrm{id}_{\mathbb{C}}ιξ⊗idC preserves the Hodge decomposition,
ιξ⊗idQℓιξ⊗idQâ„“iota_(xi)oxid_(Q_(â„“))\iota_{\xi} \otimes \operatorname{id}_{\mathbb{Q}_{\ell}}ιξ⊗idQâ„“ is Gal(Q¯/F′)Galâ¡Q¯/F′Gal( bar(Q)//F^('))\operatorname{Gal}\left(\overline{\mathbb{Q}} / F^{\prime}\right)Galâ¡(Q¯/F′)-equivariant for all ℓâ„“â„“\ellâ„“.
Our proof proceeds as follows. First, we choose an ambient variety XXXXX equipped with an embedding j:X1×X2↪Xj:X1×X2↪Xj:X_(1)xxX_(2)↪Xj: X_{1} \times X_{2} \hookrightarrow Xj:X1×X2↪X. Then we construct a class Ξ∈Hd,d(X)Ξ∈Hd,d(X)Xi inH^(d,d)(X)\Xi \in H^{d, d}(X)Ξ∈Hd,d(X) such that the (π⊗π)(π⊗π)(pi ox pi)(\pi \otimes \pi)(π⊗π)-isotypic component (j∗Ξ)π⊗πj∗Ξπ⊗π(j^(**)Xi)_(pi ox pi)\left(j^{*} \Xi\right)_{\pi \otimes \pi}(j∗Ξ)π⊗π of the pullback j∗Ξ∈Hd,d(X1×X2)j∗Ξ∈Hd,dX1×X2j^(**)Xi inH^(d,d)(X_(1)xxX_(2))j^{*} \Xi \in H^{d, d}\left(X_{1} \times X_{2}\right)j∗Ξ∈Hd,d(X1×X2) is nonzero. Finally, we modify ΞΞXi\XiΞ in such a way that ΞΞXi\XiΞ lies in H2d(X,Q)H2d(X,Q)H^(2d)(X,Q)H^{2 d}(X, \mathbb{Q})H2d(X,Q) and ξ=(j∗Ξ)π⊗πξ=j∗Ξπ⊗πxi=(j^(**)Xi)_(pi ox pi)\xi=\left(j^{*} \Xi\right)_{\pi \otimes \pi}ξ=(j∗Ξ)π⊗π is the desired Hodge cycle.
More precisely, fix a totally imaginary quadratic extension EEEEE of FFFFF which embeds into B1B1B_(1)B_{1}B1 and B2B2B_(2)B_{2}B2. For i=1,2i=1,2i=1,2i=1,2i=1,2, let Vi=BiVi=BiV_(i)=B_(i)\mathbf{V}_{i}=B_{i}Vi=Bi be the 2-dimensional Hermitian space over EEEEE such that
Then we may replace XiXiX_(i)X_{i}Xi by the Shimura variety for GU(Vi)GUViGU(V_(i))\mathrm{GU}\left(\mathbf{V}_{i}\right)GU(Vi). Consider the 4-dimensional Hermitian space V=V1⊕V2V=V1⊕V2V=V_(1)o+V_(2)\mathbf{V}=\mathbf{V}_{1} \oplus \mathbf{V}_{2}V=V1⊕V2 over EEEEE and put G=GU(V)G=GU(V)G=GU(V)\mathbf{G}=\mathrm{GU}(\mathbf{V})G=GU(V). Note that if we write v1,…,vdv1,…,vdv_(1),dots,v_(d)v_{1}, \ldots, v_{d}v1,…,vd (resp. vd+1,…,v[F:Q]vd+1,…,v[F:Q]v_(d+1),dots,v_([F:Q])v_{d+1}, \ldots, v_{[F: \mathbb{Q}]}vd+1,…,v[F:Q] ) for the infinite places of FFFFF at which B1B1B_(1)B_{1}B1 and B2B2B_(2)B_{2}B2 are split (resp. ramified), then we have
G(Fvi)={GU(2,2) if i≤dGU(4) if i>dGFvi=GUâ¡(2,2) if i≤dGUâ¡(4) if i>dG(F_(v_(i)))={[GU(2","2)," if "i <= d],[GU(4)," if "i > d]:}\mathbf{G}\left(F_{v_{i}}\right)= \begin{cases}\operatorname{GU}(2,2) & \text { if } i \leq d \\ \operatorname{GU}(4) & \text { if } i>d\end{cases}G(Fvi)={GUâ¡(2,2) if i≤dGUâ¡(4) if i>d
Put G∞=G(F⊗QR)G∞=GF⊗QRG_(oo)=G(Fox_(Q)R)\mathbf{G}_{\infty}=\mathbf{G}\left(F \otimes_{\mathbb{Q}} \mathbb{R}\right)G∞=G(F⊗QR) and let g∞g∞g_(oo)g_{\infty}g∞ denote the complexified Lie algebra of G∞G∞G_(oo)\mathbf{G}_{\infty}G∞. Let K∞K∞K_(oo)\mathbf{K}_{\infty}K∞ be the standard maximal connected compact modulo center subgroup of G∞G∞G_(oo)\mathbf{G}_{\infty}G∞. Let XXXXX be the Shimura variety for GGG\mathbf{G}G (with respect to some neat open compact subgroup KfKfK_(f)\mathbf{K}_{f}Kf of G(Af)GAfG(A_(f))\mathbf{G}\left(\mathbb{A}_{f}\right)G(Af) ), which is equipped with the embedding j:X1×X2↪Xj:X1×X2↪Xj:X_(1)xxX_(2)↪Xj: X_{1} \times X_{2} \hookrightarrow Xj:X1×X2↪X induced by the natural embedding
where the left-hand side is the subgroup of GU(V1)×GU(V2)GUâ¡V1×GUV2GU(V_(1))xxGU(V_(2))\operatorname{GU}\left(\mathbf{V}_{1}\right) \times \mathrm{GU}\left(\mathbf{V}_{2}\right)GUâ¡(V1)×GU(V2) which consists of elements with the same similitude factor. Then Matsushima's formula says that
σσsigma\sigmaσ runs over irreducible unitary representations of G(A)G(A)G(A)\mathbf{G}(\mathbb{A})G(A),
σ∞σ∞sigma_(oo)\sigma_{\infty}σ∞ and σfσfsigma_(f)\sigma_{f}σf are the infinite and finite components of σσsigma\sigmaσ, respectively,
m(σ)m(σ)m(sigma)m(\sigma)m(σ) is the multiplicity of σσsigma\sigmaσ in the automorphic discrete spectrum of GGG\mathbf{G}G,
H∗(g∞,K∞;σ∞)H∗g∞,K∞;σ∞H^(**)(g_(oo),K_(oo);sigma_(oo))H^{*}\left(g_{\infty}, \mathbf{K}_{\infty} ; \sigma_{\infty}\right)H∗(g∞,K∞;σ∞) is the relative Lie algebra cohomology,
σfKfσfKfsigma_(f)^(K_(f))\sigma_{f}^{\mathbf{K}_{f}}σfKf is the space of KfKfK_(f)\mathbf{K}_{f}Kf-fixed vectors in σfσfsigma_(f)\sigma_{f}σf.
Hence, to construct a class ΞΞXi\XiΞ as above, we need to find an irreducible automorphic representation σσsigma\sigmaσ of G(A)G(A)G(A)\mathbf{G}(\mathbb{A})G(A) which satisfies the following properties:
(a) To achieve the condition Ξ∈Hd,d(X)Ξ∈Hd,d(X)Xi inH^(d,d)(X)\Xi \in H^{d, d}(X)Ξ∈Hd,d(X), we require that
If this is the case, then it follows from the result of Vogan-Zuckerman [52] that σviσvisigma_(v_(i))\sigma_{v_{i}}σvi (restricted to U(V)(Fvi))U(V)Fvi{:U(V)(F_(v_(i))))\left.\mathrm{U}(\mathbf{V})\left(F_{v_{i}}\right)\right)U(V)(Fvi)) is equal to
{1 or Aq if i≤d1 if i>d1 or Aq if i≤d1 if i>d{[1" or "A_(q)," if "i <= d],[1," if "i > d]:}\begin{cases}\mathbf{1} \text { or } A_{\mathrm{q}} & \text { if } i \leq d \\ \mathbf{1} & \text { if } i>d\end{cases}{1 or Aq if i≤d1 if i>d
Here 111\mathbf{1}1 denotes the trivial representation and AqAqA_(q)A_{\mathfrak{q}}Aq is the cohomological representation of U(2,2)U(2,2)U(2,2)\mathrm{U}(2,2)U(2,2) associated to the θθtheta\thetaθ-stable parabolic subalgebra q with Levi component u(1,1)⊕u(1,1)u(1,1)⊕u(1,1)u(1,1)o+u(1,1)\mathfrak{u}(1,1) \oplus \mathfrak{u}(1,1)u(1,1)⊕u(1,1). We further require that σvi=Aqσvi=Aqsigma_(v_(i))=A_(q)\sigma_{v_{i}}=A_{\mathfrak{q}}σvi=Aq if i≤di≤di <= di \leq di≤d in order not to make σσsigma\sigmaσ 1-dimensional.
(b) To achieve the condition (j∗Ξ)π⊗π≠0j∗Ξπ⊗π≠0(j^(**)Xi)_(pi ox pi)!=0\left(j^{*} \Xi\right)_{\pi \otimes \pi} \neq 0(j∗Ξ)π⊗π≠0, we require the nonvanishing of the automorphic period
For (a), we use the following variant of the theta lifting from SL2SL2SL_(2)\operatorname{SL}_{2}SL2 to SO(4,2)∼U(2,2)SOâ¡(4,2)∼U(2,2)SO(4,2)∼U(2,2)\operatorname{SO}(4,2) \sim \mathrm{U}(2,2)SOâ¡(4,2)∼U(2,2) or SO(6)∼U(4)SO(6)∼U(4)SO(6)∼U(4)\mathrm{SO}(6) \sim \mathrm{U}(4)SO(6)∼U(4), where ∼∼∼\sim∼ denotes an isogeny. Let BBBBB be the quaternion algebra over FFFFF such that B=B1⋅B2B=B1â‹…B2B=B_(1)*B_(2)B=B_{1} \cdot B_{2}B=B1â‹…B2 in the Brauer group, so that BBBBB is split at all infinite places of FFFFF. We may regard V=∧2VV=∧2VV=^^^(2)VV=\wedge^{2} \mathbf{V}V=∧2V as a 3-dimensional skew-Hermitian space over BBBBB such that
Then any σσsigma\sigmaσ as in (a) with trivial central character is a theta lift of an irreducible cuspidal automorphic representation τÏ„tau\tauÏ„ of GU(W)(A)GUâ¡(W)(A)GU(W)(A)\operatorname{GU}(W)(\mathbb{A})GUâ¡(W)(A) such that τvÏ„vtau_(v)\tau_{v}Ï„v is the discrete series of weight 3 for all infinite places vvvvv of FFFFF. For (b), we can easily find the corresponding τÏ„tau\tauÏ„ by using the following seesaw diagram:
Here V=V1⊕V2V=V1⊕V2V=V_(1)o+V_(2)V=V_{1} \oplus V_{2}V=V1⊕V2 is a decomposition into 1- and 2-dimensional skew-Hermitian spaces over BBBBB such that
For simplicity, we further assume that the Hecke eigenvalues of σσsigma\sigmaσ lie in QQQ\mathbb{Q}Q. Thus we obtain a class Ξ∈Hd,d(X)Ξ∈Hd,d(X)Xi inH^(d,d)(X)\Xi \in H^{d, d}(X)Ξ∈Hd,d(X) such that ξ=(j∗Ξ)π⊗πξ=j∗Ξπ⊗πxi=(j^(**)Xi)_(pi ox pi)\xi=\left(j^{*} \Xi\right)_{\pi \otimes \pi}ξ=(j∗Ξ)π⊗π induces an isomorphism
To be precise, we need to use the theta lifting valued in cohomology developed by KudlaMillson [40,41]. On the other hand, we can determine the near equivalence class of σσsigma\sigmaσ and prove that
Hence we can modify ΞΞXi\XiΞ in such a way that ΞΞXi\XiΞ lies in H2d(X,Q)σH2d(X,Q)σH^(2d)(X,Q)_(sigma)H^{2 d}(X, \mathbb{Q})_{\sigma}H2d(X,Q)σ, so that it is a Hodge cycle. Finally, it follows from the result of Kisin-Shin-Zhu [38] that
for some positive integer mmmmm, from which Theorem 4.1 follows immediately.
Remark 4.2. It is desirable to upgrade ξξxi\xiξ to an absolute Hodge cycle in the sense of Deligne [14], but this remains an open problem.
ACKNOWLEDGMENTS
The author would like to thank Tamotsu Ikeda for his constant support and encouragement. The author would also like to thank Wee Teck Gan, Erez Lapid, and Kartik Prasanna for their generosity in sharing their ideas and numerous valuable discussions over the years.
FUNDING
This work was partially supported by JSPS KAKENHI Grant Number 19H01781.
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ATSUSHI ICHINO
Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan, ichino @ math.kyoto-u.ac.jp
Diophantine approximation, Duffin-Schaeffer conjecture, Metric Number Theory, compression arguments, density increment, graph theory
1. DIOPHANTINE APPROXIMATION
Let xxxxx be an irrational number. In many settings, practical and theoretical, it is important to find fractions a/qa/qa//qa / qa/q of small numerator and denominator that approximate it well. This fundamental question lies at the core of the field of Diophantine approximation.
1.1. First principles
The "high-school way" of approximating xxxxx is to use its decimal expansion. This approach produces fractions a/10na/10na//10^(n)a / 10^{n}a/10n such that |x−a/10n|≈10−nx−a/10n≈10−n|x-a//10^(n)|~~10^(-n)\left|x-a / 10^{n}\right| \approx 10^{-n}|x−a/10n|≈10−n typically. However, the error can be made much smaller if we allow more general denominators [14, THEOREM 2.1].
Theorem 1.1. If x∈R∖Qx∈R∖Qx inR\\Qx \in \mathbb{R} \backslash \mathbb{Q}x∈R∖Q, then |x−a/q|<q−2|x−a/q|<q−2|x-a//q| < q^(-2)|x-a / q|<q^{-2}|x−a/q|<q−2 for infinitely many pairs (a,q)∈Z×N(a,q)∈Z×N(a,q)inZxxN(a, q) \in \mathbb{Z} \times \mathbb{N}(a,q)∈Z×N.
Dirichlet (c. 1840) gave a short and clever proof of this theorem. However, his argument is nonconstructive because it uses the pigeonhole principle. This gap is filled by the theory of continued fractions (which actually precedes Dirichlet's proof).
Given any x∈R∖Qx∈R∖Qx inR\\Qx \in \mathbb{R} \backslash \mathbb{Q}x∈R∖Q, we may write x=n0+r0≈n0x=n0+r0≈n0x=n_(0)+r_(0)~~n_(0)x=n_{0}+r_{0} \approx n_{0}x=n0+r0≈n0, where n0=⌊x⌋n0=⌊x⌋n_(0)=|__ x __|n_{0}=\lfloor x\rfloorn0=⌊x⌋ is the integer part of xxxxx and r0={x}r0={x}r_(0)={x}r_{0}=\{x\}r0={x} is its fractional part. We then let n1=⌊1/r0⌋n1=1/r0n_(1)=|__1//r_(0)__|n_{1}=\left\lfloor 1 / r_{0}\right\rfloorn1=⌊1/r0⌋ and r1={1/r0}r1=1/r0r_(1)={1//r_(0)}r_{1}=\left\{1 / r_{0}\right\}r1={1/r0}, so that x=n0+1/(n1+r1)≈n0+1/n1x=n0+1/n1+r1≈n0+1/n1x=n_(0)+1//(n_(1)+r_(1))~~n_(0)+1//n_(1)x=n_{0}+1 /\left(n_{1}+r_{1}\right) \approx n_{0}+1 / n_{1}x=n0+1/(n1+r1)≈n0+1/n1. If we repeat this process j−1j−1j-1j-1j−1 more times, we find that
(1.1)x≈n0+1n1+1⋯+1nj with ni=⌊1ri−1⌋,ri={1ri−1} for i=1,…,j. (1.1)x≈n0+1n1+1⋯+1nj with ni=1ri−1,ri=1ri−1 for i=1,…,j. {:(1.1)x~~n_(0)+(1)/(n_(1)+(1)/(cdots+(1)/(n_(j))))quad" with "n_(i)=|__(1)/(r_(i-1))__|","r_(i)={(1)/(r_(i-1))}" for "i=1","dots","j". ":}\begin{equation*}
x \approx n_{0}+\frac{1}{n_{1}+\frac{1}{\cdots+\frac{1}{n_{j}}}} \quad \text { with } n_{i}=\left\lfloor\frac{1}{r_{i-1}}\right\rfloor, r_{i}=\left\{\frac{1}{r_{i-1}}\right\} \text { for } i=1, \ldots, j \text {. } \tag{1.1}
\end{equation*}(1.1)x≈n0+1n1+1⋯+1nj with ni=⌊1ri−1⌋,ri={1ri−1} for i=1,…,j.Â
If we write this fraction as aj/qjaj/qja_(j)//q_(j)a_{j} / q_{j}aj/qj in reduced form, then a calculation reveals that
When j→∞j→∞j rarr ooj \rightarrow \inftyj→∞, the right-hand side of (1.1), often denoted by [n0;n1,…,nj]n0;n1,…,nj[n_(0);n_(1),dots,n_(j)]\left[n_{0} ; n_{1}, \ldots, n_{j}\right][n0;n1,…,nj], converges to xxxxx. The resulting representation of xxxxx is called its continued fraction expansion. The quotients aj/qjaj/qja_(j)//q_(j)a_{j} / q_{j}aj/qj are called the convergents of this expansion and they have remarkable properties [17]. We list some of them below, with the first giving a constructive proof of Theorem 1.1.
Theorem 1.2. Assume the above set-up and notations.
Hence, together with Theorem 1.2(c), this implies that xxxxx is badly approximable if and only if the sequence (nj)j=0∞njj=0∞(n_(j))_(j=0)^(oo)\left(n_{j}\right)_{j=0}^{\infty}(nj)j=0∞ is bounded. Famously, Lagrange proved that the quadratic irrational numbers are in one-to-one correspondence with the continued fractions that are eventually periodic [17, $10]$10]$10]\$ 10]$10]. In particular, all such numbers are badly approximable.
A related concept to badly approximable numbers is the irrationality measure. For each x∈Rx∈Rx inRx \in \mathbb{R}x∈R, we define it to be
Instead of trying to reduce the error term in Dirichlet's approximation theorem, we often require a different type of improvement: restricting the denominators qqqqq to lie in some special set SSSSS. The theory of continued fractions is of limited use for such problems because the denominators it produces satisfy rigid recursive relations (cf. (1.2)).
For rational approximation with prime or square denominators, the best results at the moment are due to Matomäki [21] and Zaharescu [28], respectively.
Theorem 1.3 (Matomäki (2009)). Let xxxxx be an irrational number and let ε>0ε>0epsi > 0\varepsilon>0ε>0. There are infinitely many integers aaaaa and prime numbers ppppp such that |x−a/p|<p−4/3+ε|x−a/p|<p−4/3+ε|x-a//p| < p^(-4//3+epsi)|x-a / p|<p^{-4 / 3+\varepsilon}|x−a/p|<p−4/3+ε.
Theorem 1.4 (Zaharescu (1995)). Let xxxxx be an irrational number and let ε>0ε>0epsi > 0\varepsilon>0ε>0. There are infinitely many pairs (a,q)∈Z×N(a,q)∈Z×N(a,q)inZxxN(a, q) \in \mathbb{Z} \times \mathbb{N}(a,q)∈Z×N such that |x−a/q2|<q−8/3+εx−a/q2<q−8/3+ε|x-a//q^(2)| < q^(-8//3+epsi)\left|x-a / q^{2}\right|<q^{-8 / 3+\varepsilon}|x−a/q2|<q−8/3+ε.
Two important open problems are to show that Theorems 1.3 and 1.4 remain true even if we replace the constants 4/34/34//34 / 34/3 and 8/38/38//38 / 38/3 by 2 and 3, respectively.
In order to give precise meaning to the word "proportion," we shall endow RRR\mathbb{R}R with a measure. Here, we will mainly use the Lebesgue measure (denoted by "meas").
2.1. The theorems of Khinchin and JarnÃk-Besicovitch
(2.1)A:={x∈[0,1]:|x−a/q|<Δq for infinitely many pairs (a,q)∈Z×N}(2.1)A:=x∈[0,1]:|x−a/q|<Δq for infinitely many pairs (a,q)∈Z×N{:(2.1)A:={x in[0,1]:|x-a//q| < Delta_(q)" for infinitely many pairs "(a,q)inZxxN}:}\begin{equation*}
\mathcal{A}:=\left\{x \in[0,1]:|x-a / q|<\Delta_{q} \text { for infinitely many pairs }(a, q) \in \mathbb{Z} \times \mathbb{N}\right\} \tag{2.1}
\end{equation*}(2.1)A:={x∈[0,1]:|x−a/q|<Δq for infinitely many pairs (a,q)∈Z×N}
(a) If ∑q=1∞qΔq<∞∑q=1∞ qΔq<∞sum_(q=1)^(oo)qDelta_(q) < oo\sum_{q=1}^{\infty} q \Delta_{q}<\infty∑q=1∞qΔq<∞, then meas (A)=0(A)=0(A)=0(\mathcal{A})=0(A)=0.
(b) If ∑q=1∞qΔq=∞∑q=1∞ qΔq=∞sum_(q=1)^(oo)qDelta_(q)=oo\sum_{q=1}^{\infty} q \Delta_{q}=\infty∑q=1∞qΔq=∞ and the sequence (q2Δq)q=1∞q2Δqq=1∞(q^(2)Delta_(q))_(q=1)^(oo)\left(q^{2} \Delta_{q}\right)_{q=1}^{\infty}(q2Δq)q=1∞ is decreasing, then meas(A)=1measâ¡(A)=1meas(A)=1\operatorname{meas}(\mathcal{A})=1measâ¡(A)=1.
In particular, Corollary 2.2 implies that the set of badly approximable numbers has null Lebesgue measure. On the other hand, it also says that almost all real numbers have irrationality measure equal to 2 . This last result is the main motivation behind the conjecture that μ(π)=2μ(Ï€)=2mu(pi)=2\mu(\pi)=2μ(Ï€)=2 : we expect πÏ€pi\piÏ€ to behave like a "typical" real number.
In order to understand better the forces at play here, it is useful to recast Khinchin's theorem in probabilistic terms. For each qqqqq, let us define the set
Then A={x∈[0,1]:x∈AqA=x∈[0,1]:x∈AqA={x in[0,1]:x inA_(q):}\mathscr{A}=\left\{x \in[0,1]: x \in \mathscr{A}_{q}\right.A={x∈[0,1]:x∈Aq infinitely often }}}\}}, which we often write as A=limsupq→∞AqA=limsupq→∞ AqA=lims u p_(q rarr oo)A_(q)\mathscr{A}=\lim \sup _{q \rightarrow \infty} \mathscr{A}_{q}A=limsupq→∞Aq. We may thus view AAA\mathscr{A}A as the event that for a number chosen uniformly at random from [0,1][0,1][0,1][0,1][0,1], an infinite number of the events A1,A2,…A1,A2,…A_(1),A_(2),dots\mathscr{A}_{1}, \mathscr{A}_{2}, \ldotsA1,A2,… occur. A classical result from probability theory due to Borel and Cantelli [14, LEMMAS 1.2 E 1.3] studies precisely this kind of questions.
(a) (The first Borel-Cantelli lemma) If ∑j=1∞P(Ej)<∞∑j=1∞ PEj<∞sum_(j=1)^(oo)P(E_(j)) < oo\sum_{j=1}^{\infty} \mathbb{P}\left(E_{j}\right)<\infty∑j=1∞P(Ej)<∞, then P(E)=0P(E)=0P(E)=0\mathbb{P}(E)=0P(E)=0.
(b) (The second Borel-Cantelli lemma) If ∑j=1∞P(Ej)=∞∑j=1∞ PEj=∞sum_(j=1)^(oo)P(E_(j))=oo\sum_{j=1}^{\infty} \mathbb{P}\left(E_{j}\right)=\infty∑j=1∞P(Ej)=∞ and the events E1,E2,…E1,E2,…E_(1),E_(2),dotsE_{1}, E_{2}, \ldotsE1,E2,… are mutually independent, then P(E)=1P(E)=1P(E)=1\mathbb{P}(E)=1P(E)=1.
Remark. Let NNNNN be the random variable that counts how many of the events E1,E2,…E1,E2,…E_(1),E_(2),dotsE_{1}, E_{2}, \ldotsE1,E2,… occur. We have E[N]=∑j=1∞P(Ej)E[N]=∑j=1∞ PEjE[N]=sum_(j=1)^(oo)P(E_(j))\mathbb{E}[N]=\sum_{j=1}^{\infty} \mathbb{P}\left(E_{j}\right)E[N]=∑j=1∞P(Ej). Hence, Theorem 2.4 says that, under certain assumptions, N=∞N=∞N=ooN=\inftyN=∞ almost surely if, and only if, E[N]=∞E[N]=∞E[N]=oo\mathbb{E}[N]=\inftyE[N]=∞.
In particular, we see that part (a) of Khinchin's theorem is a direct consequence of the first Borel-Cantelli lemma. On the other hand, the second Borel-Cantelli lemma relies crucially on the assumption that the events EjEjE_(j)E_{j}Ej are independent of each other, something that fails generically for the events AqAqA_(q)\mathscr{A}_{q}Aq. However, there are variations of the second BorelCantelli lemma, where the assumption of independence can be replaced by weaker quasiindependence conditions on the relevant events (cf. Section 3.1). From this perspective, part (b) of Khinchin's theorem can be seen as saying that the condition that the sequence (q2Δq)q=1∞q2Δqq=1∞(q^(2)Delta_(q))_(q=1)^(oo)\left(q^{2} \Delta_{q}\right)_{q=1}^{\infty}(q2Δq)q=1∞ is decreasing guarantees enough approximate independence between the events AqAqA_(q)\mathcal{A}_{q}Aq so that the conclusion of the second Borel-Cantelli lemma remains valid.
In 1941, Duffin and Schaeffer published a seminal paper [8] that studied precisely what is the right way to generalize Khinchin's theorem so that the simple zero-one law of Borel-Cantelli holds. Their starting point was the simple observation that certain choices of the quantities ΔqΔqDelta_(q)\Delta_{q}Δq create many dependencies between the sets AqAqA_(q)\mathscr{A}_{q}Aq, thus rendering many of the denominators qqqqq redundant. Indeed, note, for example, that if Δ3=Δ15Δ3=Δ15Delta_(3)=Delta_(15)\Delta_{3}=\Delta_{15}Δ3=Δ15, then A3⊆A15A3⊆A15A_(3)subeA_(15)\mathcal{A}_{3} \subseteq \mathscr{A}_{15}A3⊆A15 because each fraction with denominator 3 can also be written as a fraction with denominator 15. By exploiting this simple idea, Duffin and Schaeffer proved the following result:
Since Aq⊆AqjAq⊆AqjA_(q)subeA_(q_(j))\mathscr{A}_{q} \subseteq \mathscr{A}_{q_{j}}Aq⊆Aqj for all q∈Sjq∈Sjq inS_(j)q \in S_{j}q∈Sj, we have A=lim supsinj→∞AqjA=lim supsinj→∞â¡AqjA=l i m   s u psin_(j rarr oo)A_(q_(j))\mathcal{A}=\limsup \sin _{j \rightarrow \infty} \mathscr{A}_{q_{j}}A=lim supsinj→∞â¡Aqj. In addition, since ∑j=1∞qjΔqj<∞∑j=1∞ qjΔqj<∞sum_(j=1)^(oo)q_(j)Delta_(q_(j)) < oo\sum_{j=1}^{\infty} q_{j} \Delta_{q_{j}}<\infty∑j=1∞qjΔqj<∞, we have meas (limsupj→∞Aqj)=0limsupj→∞ Aqj=0(lims u p_(j rarr oo)A_(q_(j)))=0\left(\lim \sup _{j \rightarrow \infty} \mathcal{A}_{q_{j}}\right)=0(limsupj→∞Aqj)=0 by Theorem 2.1(a). Hence, meas (A)=0(A)=0(A)=0(\mathcal{A})=0(A)=0, as needed. On the other hand, we have that
In order to avoid the above kind of counterexamples to the generalized Khinchin theorem, Duffin and Schaeffer were naturally led to consider a modified setup, where only reduced fractions are used as approximations. They thus defined
(2.4)A∗:={x∈[0,1]:|x−a/q|<Δq for infinitely many reduced fractions a/q}(2.4)A∗:=x∈[0,1]:|x−a/q|<Δq for infinitely many reduced fractions a/q{:(2.4)A^(**):={x in[0,1]:|x-a//q| < Delta_(q)" for infinitely many reduced fractions "a//q}:}\begin{equation*}
\mathcal{A}^{*}:=\left\{x \in[0,1]:|x-a / q|<\Delta_{q} \text { for infinitely many reduced fractions } a / q\right\} \tag{2.4}
\end{equation*}(2.4)A∗:={x∈[0,1]:|x−a/q|<Δq for infinitely many reduced fractions a/q}
We may write A∗A∗A^(**)\mathcal{A}^{*}A∗ as the lim sup of the sets
is Euler's totient function. They then conjectured that the sets Aq∗Aq∗A_(q)^(**)\mathscr{A}_{q}^{*}Aq∗ have enough mutual quasiindependence so that a simple zero-one law holds, as per the Borel-Cantelli lemmas.
(a) If ∑q=1∞φ(q)Δq<∞∑q=1∞ φ(q)Δq<∞sum_(q=1)^(oo)varphi(q)Delta_(q) < oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}<\infty∑q=1∞φ(q)Δq<∞, then meas (A∗)=0A∗=0(A^(**))=0\left(\mathscr{A}^{*}\right)=0(A∗)=0.
(b) If ∑q=1∞φ(q)Δq=∞∑q=1∞ φ(q)Δq=∞sum_(q=1)^(oo)varphi(q)Delta_(q)=oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}=\infty∑q=1∞φ(q)Δq=∞, then meas (A∗)=1A∗=1(A^(**))=1\left(\mathscr{A}^{*}\right)=1(A∗)=1.
Of course, part (a) follows from Theorem 2.4(a); the main difficulty is to prove (b).
The Duffin-Schaeffer conjecture is strikingly simple and general. Nonetheless, it does not answer our original question: what is the correct generalization of Khinchin's theorem, where we may use nonreduced fractions? This gap was filled by Catlin [7].
(a) If ∑q=1∞φ(q)Δq′<∞∑q=1∞ φ(q)Δq′<∞sum_(q=1)^(oo)varphi(q)Delta_(q)^(') < oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}^{\prime}<\infty∑q=1∞φ(q)Δq′<∞, then meas(A)=0measâ¡(A)=0meas(A)=0\operatorname{meas}(\mathcal{A})=0measâ¡(A)=0.
(b) If ∑q=1∞φ(q)Δq′=∞∑q=1∞ φ(q)Δq′=∞sum_(q=1)^(oo)varphi(q)Delta_(q)^(')=oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}^{\prime}=\infty∑q=1∞φ(q)Δq′=∞, then meas (A)=1(A)=1(A)=1(\mathcal{A})=1(A)=1.
As Catlin noticed, his conjecture is a direct corollary of that by Duffin and Schaeffer. Indeed, let us consider the set
A′={x∈[0,1]:|x−a/q|<Δq′ for infinitely many reduced fractions a/q}A′=x∈[0,1]:|x−a/q|<Δq′ for infinitely many reduced fractions a/qA^(')={x in[0,1]:|x-a//q| < Delta_(q)^(')" for infinitely many reduced fractions "a//q}\mathcal{A}^{\prime}=\left\{x \in[0,1]:|x-a / q|<\Delta_{q}^{\prime} \text { for infinitely many reduced fractions } a / q\right\}A′={x∈[0,1]:|x−a/q|<Δq′ for infinitely many reduced fractions a/q}
This is the set A∗A∗A^(**)\mathscr{A}^{*}A∗ with the quantities ΔqΔqDelta_(q)\Delta_{q}Δq replaced by Δq′Δq′Delta_(q)^(')\Delta_{q}^{\prime}Δq′, so we may apply the DuffinSchaeffer conjecture to it. In addition, it is straightforward to check that
when Δq→0Δq→0Delta_(q)rarr0\Delta_{q} \rightarrow 0Δq→0. This settles Catlin's conjecture in this case. On the other hand, if Δq↛0Δq↛0Delta_(q)↛0\Delta_{q} \nrightarrow 0Δq↛0, then A=[0,1]A=[0,1]A=[0,1]\mathcal{A}=[0,1]A=[0,1] and ∑q=1∞φ(q)Δq′=∞∑q=1∞ φ(q)Δq′=∞sum_(q=1)^(oo)varphi(q)Delta_(q)^(')=oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}^{\prime}=\infty∑q=1∞φ(q)Δq′=∞, so that Catlin's conjecture is trivially true.
Just like in Theorem 2.3 of JarnÃk and Besicovitch, it would be important to also have information about the Hausdorff dimension of the sets AAA\mathcal{A}A and A∗A∗A^(**)\mathscr{A}^{*}A∗ in the case when they have null Lebesgue measure. In light of relation (2.6), it suffices to answer this question for the latter set. Beresnevich and Velani [4] proved the remarkable result that the DuffinSchaeffer conjecture implies a Hausdorff measure version of itself. This is a consequence of a much more general Mass Transfer Principle that they established, and which allows transfering information concerning the Lebesgue measure of certain lim sup sets to the Hausdorff measure of rescaled versions of them. As a corollary, they proved:
2.3. Progress towards the Duffin-Schaeffer conjecture
Since its introduction in 1941, the Duffin-Schaeffer conjectured has been the subject of intensive research activity, with various special cases proven over the years. This process came to a conclusion recently with the proof of the full conjecture [20].
Theorem 2.7 (Koukoulopoulos-Maynard (2020)). The Duffin-Schaeffer conjecture is true.
We will outline the main ideas of the proof of Theorem 2.7 in Section 3. But first we give an account of the work that preceeded it.
In the same paper where they introduced their conjecture, Duffin and Schaeffer proved the first general case of it:
Theorem 2.8 (Duffin-Schaeffer (1941)). The Duffin-Schaeffer conjecture is true for all sequences (Δq)q=1∞Δqq=1∞(Delta_(q))_(q=1)^(oo)\left(\Delta_{q}\right)_{q=1}^{\infty}(Δq)q=1∞ such that
To appreciate this result, we must make a few comments about condition (2.7). Note that its left-hand side is the average value of φ(q)/qφ(q)/qvarphi(q)//q\varphi(q) / qφ(q)/q over q∈[1,Q]q∈[1,Q]q in[1,Q]q \in[1, Q]q∈[1,Q], where qqqqq is weighted by wq:=qΔqwq:=qΔqw_(q):=qDelta_(q)w_{q}:=q \Delta_{q}wq:=qΔq. In particular, we may restrict our attention to qqqqq with Δq>0Δq>0Delta_(q) > 0\Delta_{q}>0Δq>0. Now, we know
for j=1,2,…,1+⌊0.01loglogq⌋j=1,2,…,1+⌊0.01logâ¡logâ¡q⌋j=1,2,dots,1+|__0.01 log log q __|j=1,2, \ldots, 1+\lfloor 0.01 \log \log q\rfloorj=1,2,…,1+⌊0.01logâ¡logâ¡q⌋. We would then have
by Mertens' theorem [19, THEOREM 3.4]. This is much smaller than ej/j2ej/j2e^(j)//j^(2)e^{j} / j^{2}ej/j2, so (2.10) should fail rarely as j→∞j→∞j rarr ooj \rightarrow \inftyj→∞. (For instance, we may use Markov's inequality to see this claim.)
In conclusion, Theorem 2.8 settles the Duffin-Schaeffer conjecture when ΔqΔqDelta_(q)\Delta_{q}Δq is mainly supported on "normal" integers, without too many small prime factors. In particular, it implies a significant improvement of Theorems 1.3 and 1.4 for almost all x∈Rx∈Rx inRx \in \mathbb{R}x∈R.
Corollary 2.9. For almost all x∈Rx∈Rx inRx \in \mathbb{R}x∈R, there are infinitely many reduced fractions a/pa/pa//pa / pa/p and b/q2b/q2b//q^(2)b / q^{2}b/q2 such that ppppp is prime, |x−a/p|<p−2|x−a/p|<p−2|x-a//p| < p^(-2)|x-a / p|<p^{-2}|x−a/p|<p−2 and |x−b/q2|<q−3x−b/q2<q−3|x-b//q^(2)| < q^(-3)\left|x-b / q^{2}\right|<q^{-3}|x−b/q2|<q−3.
The next important step towards the Duffin-Schaeffer conjecture is a remarkable zero-one law due to Gallagher [12].
Theorem 2.10 (Gallagher (1961)). If A∗A∗A^(**)\mathcal{A}^{*}A∗ is as in (2.4), then meas (A∗)∈{0,1}A∗∈{0,1}(A^(**))in{0,1}\left(\mathcal{A}^{*}\right) \in\{0,1\}(A∗)∈{0,1}.
Gallagher's theorem says grosso modo that either we chose the quantities ΔqΔqDelta_(q)\Delta_{q}Δq to be "too small" and thus missed almost all real numbers, or we chose them "sufficiently large" so that almost all numbers have the desired rational approximations. The Duffin-Schaeffer conjecture is then the simplest possible criterion to decide in which case we are.
The proof of Theorem 2.10 is a clever adaptation of an ergodic-theoretic argument due to Cassels [6] in the simpler setting of nonreduced rational approximations. We give Cassel's proof and refer the interested readers to [12,14][12,14][12,14][12,14][12,14] for the proof of Theorem 2.10.
Theorem 2.11 (Cassels (1950)). If AAA\mathscr{A}A is as in (2.1), then meas(A) )∈{0,1})∈{0,1})in{0,1}) \in\{0,1\})∈{0,1}.
The first significant step towards establishing the Duffin-Schaeffer conjecture for irregular sequences ΔqΔqDelta_(q)\Delta_{q}Δq, potentially supported on integers with lots of small prime factors, was carried out by ErdÅ‘s [10] and Vaaler [27].
Theorem 2.12 (ErdÅ‘s (1970), Vaaler (1978)). The Duffin-Schaeffer conjecture is true for all sequences (Δq)q=1∞Δqq=1∞(Delta_(q))_(q=1)^(oo)\left(\Delta_{q}\right)_{q=1}^{\infty}(Δq)q=1∞ such that Δq=O(1/q2)Δq=O1/q2Delta_(q)=O(1//q^(2))\Delta_{q}=O\left(1 / q^{2}\right)Δq=O(1/q2) for all qqqqq.
Following this result, a lot of research focused on proving the Duffin-Schaeffer conjecture when the series ∑q=1∞φ(q)Δq∑q=1∞ φ(q)Δqsum_(q=1)^(oo)varphi(q)Delta_(q)\sum_{q=1}^{\infty} \varphi(q) \Delta_{q}∑q=1∞φ(q)Δq diverges fast enough (see, e.g., [14, THEOREM 3.7(III)], [3,15]). Aistleitner, Lachmann, Munsch, Technau, and Zafeiropoulos [2] proved the Duffin-
Schaeffer conjecture when ∑q=1∞φ(q)Δq/(logq)ε=∞∑q=1∞ φ(q)Δq/(logâ¡q)ε=∞sum_(q=1)^(oo)varphi(q)Delta_(q)//(log q)^(epsi)=oo\sum_{q=1}^{\infty} \varphi(q) \Delta_{q} /(\log q)^{\varepsilon}=\infty∑q=1∞φ(q)Δq/(logâ¡q)ε=∞ for some ε>0ε>0epsi > 0\varepsilon>0ε>0. A report by Aistleitner [1], announced at the same time as [20], explains how to replace (logq)ε(logâ¡q)ε(log q)^(epsi)(\log q)^{\varepsilon}(logâ¡q)ε by (loglogq)ε(logâ¡logâ¡q)ε(log log q)^(epsi)(\log \log q)^{\varepsilon}(logâ¡logâ¡q)ε.
3. THE MAIN INGREDIENTS OF THE PROOF OF THE DUFFIN-SCHAEFFER CONJECTURE
We will only use the events Aq∗Aq∗A_(q)^(**)\mathscr{A}_{q}^{*}Aq∗ with q∈[Q,R]q∈[Q,R]q in[Q,R]q \in[Q, R]q∈[Q,R]. We trivially have the union bound
If we can show that the sets Aq∗Aq∗A_(q)^(**)\mathscr{A}_{q}^{*}Aq∗ with q∈[Q,R]q∈[Q,R]q in[Q,R]q \in[Q, R]q∈[Q,R] do not overlap too much, so that
Proof. Let N=∑j1EjN=∑j 1EjN=sum_(j)1_(E_(j))N=\sum_{j} 1_{E_{j}}N=∑j1Ej. We then have E[N]=∑jP(Ej)E[N]=∑j PEjE[N]=sum_(j)P(E_(j))\mathbb{E}[N]=\sum_{j} \mathbb{P}\left(E_{j}\right)E[N]=∑jP(Ej). On the other hand, the CauchySchwarz inequality implies that
(b) If there are infinitely many disjoint intervals [Q,R][Q,R][Q,R][Q, R][Q,R] satisfying (3.3) with the same constant C>0C>0C > 0C>0C>0, then meas( limsupq→∞Aq∗)=1limsupq→∞ Aq∗=1{: lims u p_(q rarr oo)A_(q)^(**))=1\left.\lim \sup _{q \rightarrow \infty} \mathscr{A}_{q}^{*}\right)=1limsupq→∞Aq∗)=1.
3.2. A bound on the pairwise correlations
As per Proposition 3.2, we need to control the correlations of the events Aq∗Aq∗A_(q)^(**)\mathcal{A}_{q}^{*}Aq∗. To this end, we have a lemma of Pollington-Vaughan [22] (see also [10, 27]).